/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty Quasi decreasingness of the given CTRS could not be shown: (0) CTRS (1) CTRSToQTRSProof [SOUND, 0 ms] (2) QTRS (3) DependencyPairsProof [EQUIVALENT, 0 ms] (4) QDP (5) DependencyGraphProof [EQUIVALENT, 0 ms] (6) QDP (7) UsableRulesReductionPairsProof [EQUIVALENT, 0 ms] (8) QDP (9) TransformationProof [EQUIVALENT, 0 ms] (10) QDP (11) TransformationProof [EQUIVALENT, 0 ms] (12) QDP (13) TransformationProof [EQUIVALENT, 0 ms] (14) QDP (15) TransformationProof [EQUIVALENT, 0 ms] (16) QDP (17) TransformationProof [EQUIVALENT, 0 ms] (18) QDP (19) TransformationProof [EQUIVALENT, 0 ms] (20) QDP (21) TransformationProof [EQUIVALENT, 0 ms] (22) QDP (23) TransformationProof [EQUIVALENT, 0 ms] (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) TransformationProof [EQUIVALENT, 0 ms] (30) QDP (31) TransformationProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) TransformationProof [EQUIVALENT, 0 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) TransformationProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 3 ms] (44) QDP (45) TransformationProof [EQUIVALENT, 0 ms] (46) QDP (47) TransformationProof [EQUIVALENT, 0 ms] (48) QDP (49) TransformationProof [EQUIVALENT, 0 ms] (50) QDP (51) TransformationProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) TransformationProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) TransformationProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 0 ms] (64) QDP (65) TransformationProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 0 ms] (70) QDP (71) TransformationProof [EQUIVALENT, 0 ms] (72) QDP (73) TransformationProof [EQUIVALENT, 0 ms] (74) QDP (75) TransformationProof [EQUIVALENT, 0 ms] (76) QDP (77) TransformationProof [EQUIVALENT, 0 ms] (78) QDP (79) TransformationProof [EQUIVALENT, 0 ms] (80) QDP (81) TransformationProof [EQUIVALENT, 0 ms] (82) QDP (83) TransformationProof [EQUIVALENT, 0 ms] (84) QDP (85) TransformationProof [EQUIVALENT, 0 ms] (86) QDP (87) TransformationProof [EQUIVALENT, 0 ms] (88) QDP (89) TransformationProof [EQUIVALENT, 0 ms] (90) QDP (91) TransformationProof [EQUIVALENT, 0 ms] (92) QDP (93) TransformationProof [EQUIVALENT, 0 ms] (94) QDP (95) TransformationProof [EQUIVALENT, 0 ms] (96) QDP (97) TransformationProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 0 ms] (100) QDP (101) TransformationProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 1 ms] (104) QDP (105) TransformationProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) TransformationProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 0 ms] (112) QDP (113) TransformationProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 0 ms] (116) QDP (117) TransformationProof [EQUIVALENT, 0 ms] (118) QDP (119) TransformationProof [EQUIVALENT, 0 ms] (120) QDP (121) TransformationProof [EQUIVALENT, 0 ms] (122) QDP (123) TransformationProof [EQUIVALENT, 0 ms] (124) QDP (125) TransformationProof [EQUIVALENT, 0 ms] (126) QDP (127) TransformationProof [EQUIVALENT, 0 ms] (128) QDP (129) TransformationProof [EQUIVALENT, 0 ms] (130) QDP (131) TransformationProof [EQUIVALENT, 0 ms] (132) QDP (133) TransformationProof [EQUIVALENT, 0 ms] (134) QDP (135) TransformationProof [EQUIVALENT, 0 ms] (136) QDP (137) TransformationProof [EQUIVALENT, 0 ms] (138) QDP (139) TransformationProof [EQUIVALENT, 0 ms] (140) QDP (141) TransformationProof [EQUIVALENT, 0 ms] (142) QDP (143) TransformationProof [EQUIVALENT, 0 ms] (144) QDP (145) TransformationProof [EQUIVALENT, 0 ms] (146) QDP (147) TransformationProof [EQUIVALENT, 0 ms] (148) QDP (149) TransformationProof [EQUIVALENT, 0 ms] (150) QDP (151) TransformationProof [EQUIVALENT, 0 ms] (152) QDP (153) TransformationProof [EQUIVALENT, 0 ms] (154) QDP (155) TransformationProof [EQUIVALENT, 0 ms] (156) QDP (157) TransformationProof [EQUIVALENT, 0 ms] (158) QDP (159) TransformationProof [EQUIVALENT, 0 ms] (160) QDP (161) TransformationProof [EQUIVALENT, 0 ms] (162) QDP (163) TransformationProof [EQUIVALENT, 0 ms] (164) QDP (165) TransformationProof [EQUIVALENT, 0 ms] (166) QDP (167) TransformationProof [EQUIVALENT, 1 ms] (168) QDP (169) TransformationProof [EQUIVALENT, 0 ms] (170) QDP (171) TransformationProof [EQUIVALENT, 0 ms] (172) QDP (173) TransformationProof [EQUIVALENT, 0 ms] (174) QDP (175) TransformationProof [EQUIVALENT, 0 ms] (176) QDP (177) TransformationProof [EQUIVALENT, 0 ms] (178) QDP (179) TransformationProof [EQUIVALENT, 0 ms] (180) QDP (181) TransformationProof [EQUIVALENT, 0 ms] (182) QDP (183) TransformationProof [EQUIVALENT, 0 ms] (184) QDP (185) TransformationProof [EQUIVALENT, 0 ms] (186) QDP (187) TransformationProof [EQUIVALENT, 0 ms] (188) QDP (189) TransformationProof [EQUIVALENT, 0 ms] (190) QDP (191) TransformationProof [EQUIVALENT, 0 ms] (192) QDP (193) TransformationProof [EQUIVALENT, 0 ms] (194) QDP (195) TransformationProof [EQUIVALENT, 0 ms] (196) QDP (197) TransformationProof [EQUIVALENT, 0 ms] (198) QDP (199) TransformationProof [EQUIVALENT, 0 ms] (200) QDP (201) TransformationProof [EQUIVALENT, 0 ms] (202) QDP (203) TransformationProof [EQUIVALENT, 0 ms] (204) QDP (205) TransformationProof [EQUIVALENT, 0 ms] (206) QDP (207) TransformationProof [EQUIVALENT, 0 ms] (208) QDP (209) TransformationProof [EQUIVALENT, 0 ms] (210) QDP (211) TransformationProof [EQUIVALENT, 0 ms] (212) QDP (213) TransformationProof [EQUIVALENT, 0 ms] (214) QDP (215) TransformationProof [EQUIVALENT, 0 ms] (216) QDP (217) TransformationProof [EQUIVALENT, 0 ms] (218) QDP (219) TransformationProof [EQUIVALENT, 0 ms] (220) QDP (221) TransformationProof [EQUIVALENT, 0 ms] (222) QDP (223) TransformationProof [EQUIVALENT, 0 ms] (224) QDP (225) TransformationProof [EQUIVALENT, 0 ms] (226) QDP (227) TransformationProof [EQUIVALENT, 0 ms] (228) QDP (229) TransformationProof [EQUIVALENT, 0 ms] (230) QDP (231) TransformationProof [EQUIVALENT, 0 ms] (232) QDP (233) TransformationProof [EQUIVALENT, 0 ms] (234) QDP (235) TransformationProof [EQUIVALENT, 0 ms] (236) QDP (237) TransformationProof [EQUIVALENT, 0 ms] (238) QDP (239) TransformationProof [EQUIVALENT, 0 ms] (240) QDP (241) TransformationProof [EQUIVALENT, 0 ms] (242) QDP (243) TransformationProof [EQUIVALENT, 1 ms] (244) QDP (245) TransformationProof [EQUIVALENT, 0 ms] (246) QDP (247) TransformationProof [EQUIVALENT, 0 ms] (248) QDP (249) TransformationProof [EQUIVALENT, 0 ms] (250) QDP (251) TransformationProof [EQUIVALENT, 0 ms] (252) QDP (253) TransformationProof [EQUIVALENT, 0 ms] (254) QDP (255) TransformationProof [EQUIVALENT, 0 ms] (256) QDP (257) TransformationProof [EQUIVALENT, 0 ms] (258) QDP (259) TransformationProof [EQUIVALENT, 0 ms] (260) QDP (261) TransformationProof [EQUIVALENT, 0 ms] (262) QDP (263) TransformationProof [EQUIVALENT, 0 ms] (264) QDP (265) TransformationProof [EQUIVALENT, 0 ms] (266) QDP (267) TransformationProof [EQUIVALENT, 0 ms] (268) QDP (269) TransformationProof [EQUIVALENT, 0 ms] (270) QDP (271) TransformationProof [EQUIVALENT, 0 ms] (272) QDP (273) TransformationProof [EQUIVALENT, 0 ms] (274) QDP (275) TransformationProof [EQUIVALENT, 0 ms] (276) QDP (277) TransformationProof [EQUIVALENT, 0 ms] (278) QDP (279) TransformationProof [EQUIVALENT, 0 ms] (280) QDP (281) TransformationProof [EQUIVALENT, 0 ms] (282) QDP (283) TransformationProof [EQUIVALENT, 0 ms] (284) QDP (285) TransformationProof [EQUIVALENT, 0 ms] (286) QDP (287) TransformationProof [EQUIVALENT, 0 ms] (288) QDP (289) TransformationProof [EQUIVALENT, 0 ms] (290) QDP (291) TransformationProof [EQUIVALENT, 0 ms] (292) QDP (293) TransformationProof [EQUIVALENT, 0 ms] (294) QDP (295) TransformationProof [EQUIVALENT, 0 ms] (296) QDP (297) TransformationProof [EQUIVALENT, 0 ms] (298) QDP (299) TransformationProof [EQUIVALENT, 0 ms] (300) QDP (301) TransformationProof [EQUIVALENT, 0 ms] (302) QDP (303) TransformationProof [EQUIVALENT, 0 ms] (304) QDP (305) TransformationProof [EQUIVALENT, 0 ms] (306) QDP (307) TransformationProof [EQUIVALENT, 0 ms] (308) QDP (309) TransformationProof [EQUIVALENT, 0 ms] (310) QDP (311) TransformationProof [EQUIVALENT, 0 ms] (312) QDP (313) TransformationProof [EQUIVALENT, 0 ms] (314) QDP (315) TransformationProof [EQUIVALENT, 0 ms] (316) QDP (317) TransformationProof [EQUIVALENT, 0 ms] (318) QDP (319) TransformationProof [EQUIVALENT, 2 ms] (320) QDP (321) TransformationProof [EQUIVALENT, 0 ms] (322) QDP (323) TransformationProof [EQUIVALENT, 0 ms] (324) QDP (325) TransformationProof [EQUIVALENT, 0 ms] (326) QDP (327) TransformationProof [EQUIVALENT, 0 ms] (328) QDP (329) TransformationProof [EQUIVALENT, 0 ms] (330) QDP (331) TransformationProof [EQUIVALENT, 0 ms] (332) QDP (333) TransformationProof [EQUIVALENT, 0 ms] (334) QDP (335) TransformationProof [EQUIVALENT, 0 ms] (336) QDP (337) TransformationProof [EQUIVALENT, 0 ms] (338) QDP (339) TransformationProof [EQUIVALENT, 0 ms] (340) QDP (341) TransformationProof [EQUIVALENT, 0 ms] (342) QDP (343) TransformationProof [EQUIVALENT, 0 ms] (344) QDP (345) TransformationProof [EQUIVALENT, 0 ms] (346) QDP (347) TransformationProof [EQUIVALENT, 0 ms] (348) QDP (349) TransformationProof [EQUIVALENT, 0 ms] (350) QDP (351) TransformationProof [EQUIVALENT, 0 ms] (352) QDP (353) TransformationProof [EQUIVALENT, 0 ms] (354) QDP (355) TransformationProof [EQUIVALENT, 0 ms] (356) QDP (357) TransformationProof [EQUIVALENT, 0 ms] (358) QDP (359) TransformationProof [EQUIVALENT, 0 ms] (360) QDP (361) TransformationProof [EQUIVALENT, 0 ms] (362) QDP (363) TransformationProof [EQUIVALENT, 0 ms] (364) QDP (365) TransformationProof [EQUIVALENT, 0 ms] (366) QDP (367) TransformationProof [EQUIVALENT, 0 ms] (368) QDP (369) TransformationProof [EQUIVALENT, 0 ms] (370) QDP (371) TransformationProof [EQUIVALENT, 0 ms] (372) QDP (373) TransformationProof [EQUIVALENT, 0 ms] (374) QDP (375) TransformationProof [EQUIVALENT, 0 ms] (376) QDP (377) TransformationProof [EQUIVALENT, 0 ms] (378) QDP (379) TransformationProof [EQUIVALENT, 0 ms] (380) QDP (381) TransformationProof [EQUIVALENT, 0 ms] (382) QDP (383) TransformationProof [EQUIVALENT, 0 ms] (384) QDP (385) TransformationProof [EQUIVALENT, 0 ms] (386) QDP (387) TransformationProof [EQUIVALENT, 0 ms] (388) QDP (389) TransformationProof [EQUIVALENT, 0 ms] (390) QDP (391) TransformationProof [EQUIVALENT, 0 ms] (392) QDP (393) TransformationProof [EQUIVALENT, 0 ms] (394) QDP (395) TransformationProof [EQUIVALENT, 0 ms] (396) QDP (397) TransformationProof [EQUIVALENT, 0 ms] (398) QDP (399) TransformationProof [EQUIVALENT, 0 ms] (400) QDP (401) TransformationProof [EQUIVALENT, 0 ms] (402) QDP (403) TransformationProof [EQUIVALENT, 0 ms] (404) QDP (405) TransformationProof [EQUIVALENT, 0 ms] (406) QDP (407) TransformationProof [EQUIVALENT, 0 ms] (408) QDP (409) TransformationProof [EQUIVALENT, 0 ms] (410) QDP (411) TransformationProof [EQUIVALENT, 0 ms] (412) QDP (413) TransformationProof [EQUIVALENT, 0 ms] (414) QDP (415) TransformationProof [EQUIVALENT, 0 ms] (416) QDP (417) TransformationProof [EQUIVALENT, 0 ms] (418) QDP (419) TransformationProof [EQUIVALENT, 0 ms] (420) QDP (421) TransformationProof [EQUIVALENT, 0 ms] (422) QDP (423) TransformationProof [EQUIVALENT, 0 ms] (424) QDP (425) TransformationProof [EQUIVALENT, 0 ms] (426) QDP (427) TransformationProof [EQUIVALENT, 0 ms] (428) QDP (429) TransformationProof [EQUIVALENT, 0 ms] (430) QDP (431) TransformationProof [EQUIVALENT, 0 ms] (432) QDP (433) TransformationProof [EQUIVALENT, 0 ms] (434) QDP (435) TransformationProof [EQUIVALENT, 0 ms] (436) QDP (437) TransformationProof [EQUIVALENT, 0 ms] (438) QDP (439) TransformationProof [EQUIVALENT, 0 ms] (440) QDP (441) TransformationProof [EQUIVALENT, 0 ms] (442) QDP (443) TransformationProof [EQUIVALENT, 0 ms] (444) QDP (445) TransformationProof [EQUIVALENT, 0 ms] (446) QDP (447) TransformationProof [EQUIVALENT, 0 ms] (448) QDP (449) TransformationProof [EQUIVALENT, 0 ms] (450) QDP (451) TransformationProof [EQUIVALENT, 0 ms] (452) QDP (453) TransformationProof [EQUIVALENT, 0 ms] (454) QDP (455) TransformationProof [EQUIVALENT, 0 ms] (456) QDP (457) TransformationProof [EQUIVALENT, 0 ms] (458) QDP (459) TransformationProof [EQUIVALENT, 0 ms] (460) QDP (461) TransformationProof [EQUIVALENT, 0 ms] (462) QDP (463) TransformationProof [EQUIVALENT, 0 ms] (464) QDP (465) TransformationProof [EQUIVALENT, 0 ms] (466) QDP (467) TransformationProof [EQUIVALENT, 2 ms] (468) QDP (469) TransformationProof [EQUIVALENT, 0 ms] (470) QDP (471) TransformationProof [EQUIVALENT, 0 ms] (472) QDP (473) TransformationProof [EQUIVALENT, 0 ms] (474) QDP (475) TransformationProof [EQUIVALENT, 0 ms] (476) QDP (477) TransformationProof [EQUIVALENT, 0 ms] (478) QDP (479) TransformationProof [EQUIVALENT, 0 ms] (480) QDP (481) TransformationProof [EQUIVALENT, 0 ms] (482) QDP (483) TransformationProof [EQUIVALENT, 0 ms] (484) QDP (485) TransformationProof [EQUIVALENT, 0 ms] (486) QDP (487) TransformationProof [EQUIVALENT, 0 ms] (488) QDP (489) TransformationProof [EQUIVALENT, 0 ms] (490) QDP (491) TransformationProof [EQUIVALENT, 0 ms] (492) QDP (493) TransformationProof [EQUIVALENT, 0 ms] (494) QDP (495) TransformationProof [EQUIVALENT, 0 ms] (496) QDP (497) TransformationProof [EQUIVALENT, 0 ms] (498) QDP (499) TransformationProof [EQUIVALENT, 0 ms] (500) QDP (501) TransformationProof [EQUIVALENT, 0 ms] (502) QDP (503) TransformationProof [EQUIVALENT, 0 ms] (504) QDP (505) TransformationProof [EQUIVALENT, 0 ms] (506) QDP (507) TransformationProof [EQUIVALENT, 0 ms] (508) QDP (509) TransformationProof [EQUIVALENT, 0 ms] (510) QDP (511) TransformationProof [EQUIVALENT, 0 ms] (512) QDP (513) TransformationProof [EQUIVALENT, 0 ms] (514) QDP (515) TransformationProof [EQUIVALENT, 0 ms] (516) QDP (517) TransformationProof [EQUIVALENT, 0 ms] (518) QDP (519) TransformationProof [EQUIVALENT, 0 ms] (520) QDP (521) TransformationProof [EQUIVALENT, 0 ms] (522) QDP (523) TransformationProof [EQUIVALENT, 0 ms] (524) QDP (525) TransformationProof [EQUIVALENT, 0 ms] (526) QDP (527) TransformationProof [EQUIVALENT, 0 ms] (528) QDP (529) TransformationProof [EQUIVALENT, 0 ms] (530) QDP (531) TransformationProof [EQUIVALENT, 0 ms] (532) QDP (533) TransformationProof [EQUIVALENT, 0 ms] (534) QDP (535) TransformationProof [EQUIVALENT, 0 ms] (536) QDP (537) TransformationProof [EQUIVALENT, 0 ms] (538) QDP (539) TransformationProof [EQUIVALENT, 0 ms] (540) QDP (541) TransformationProof [EQUIVALENT, 0 ms] (542) QDP (543) TransformationProof [EQUIVALENT, 0 ms] (544) QDP (545) TransformationProof [EQUIVALENT, 0 ms] (546) QDP (547) TransformationProof [EQUIVALENT, 0 ms] (548) QDP (549) TransformationProof [EQUIVALENT, 0 ms] (550) QDP (551) TransformationProof [EQUIVALENT, 0 ms] (552) QDP (553) TransformationProof [EQUIVALENT, 0 ms] (554) QDP (555) TransformationProof [EQUIVALENT, 0 ms] (556) QDP (557) TransformationProof [EQUIVALENT, 0 ms] (558) QDP (559) TransformationProof [EQUIVALENT, 0 ms] (560) QDP (561) TransformationProof [EQUIVALENT, 0 ms] (562) QDP (563) TransformationProof [EQUIVALENT, 0 ms] (564) QDP (565) TransformationProof [EQUIVALENT, 0 ms] (566) QDP (567) TransformationProof [EQUIVALENT, 0 ms] (568) QDP (569) TransformationProof [EQUIVALENT, 0 ms] (570) QDP (571) TransformationProof [EQUIVALENT, 0 ms] (572) QDP (573) TransformationProof [EQUIVALENT, 0 ms] (574) QDP (575) TransformationProof [EQUIVALENT, 0 ms] (576) QDP (577) TransformationProof [EQUIVALENT, 0 ms] (578) QDP (579) TransformationProof [EQUIVALENT, 0 ms] (580) QDP (581) TransformationProof [EQUIVALENT, 0 ms] (582) QDP (583) TransformationProof [EQUIVALENT, 0 ms] (584) QDP (585) TransformationProof [EQUIVALENT, 0 ms] (586) QDP (587) TransformationProof [EQUIVALENT, 0 ms] (588) QDP (589) TransformationProof [EQUIVALENT, 0 ms] (590) QDP (591) TransformationProof [EQUIVALENT, 0 ms] (592) QDP (593) TransformationProof [EQUIVALENT, 0 ms] (594) QDP (595) TransformationProof [EQUIVALENT, 1 ms] (596) QDP (597) TransformationProof [EQUIVALENT, 0 ms] (598) QDP (599) TransformationProof [EQUIVALENT, 0 ms] (600) QDP (601) TransformationProof [EQUIVALENT, 0 ms] (602) QDP (603) TransformationProof [EQUIVALENT, 0 ms] (604) QDP (605) TransformationProof [EQUIVALENT, 0 ms] (606) QDP (607) TransformationProof [EQUIVALENT, 0 ms] (608) QDP (609) TransformationProof [EQUIVALENT, 0 ms] (610) QDP (611) TransformationProof [EQUIVALENT, 0 ms] (612) QDP (613) TransformationProof [EQUIVALENT, 0 ms] (614) QDP (615) TransformationProof [EQUIVALENT, 0 ms] (616) QDP (617) TransformationProof [EQUIVALENT, 0 ms] (618) QDP (619) TransformationProof [EQUIVALENT, 0 ms] (620) QDP (621) TransformationProof [EQUIVALENT, 0 ms] (622) QDP (623) TransformationProof [EQUIVALENT, 0 ms] (624) QDP (625) TransformationProof [EQUIVALENT, 0 ms] (626) QDP (627) TransformationProof [EQUIVALENT, 0 ms] (628) QDP (629) TransformationProof [EQUIVALENT, 0 ms] (630) QDP (631) TransformationProof [EQUIVALENT, 0 ms] (632) QDP (633) TransformationProof [EQUIVALENT, 0 ms] (634) QDP (635) TransformationProof [EQUIVALENT, 0 ms] (636) QDP (637) TransformationProof [EQUIVALENT, 0 ms] (638) QDP (639) TransformationProof [EQUIVALENT, 0 ms] (640) QDP (641) TransformationProof [EQUIVALENT, 0 ms] (642) QDP (643) TransformationProof [EQUIVALENT, 0 ms] (644) QDP (645) TransformationProof [EQUIVALENT, 0 ms] (646) QDP (647) TransformationProof [EQUIVALENT, 0 ms] (648) QDP (649) TransformationProof [EQUIVALENT, 0 ms] (650) QDP (651) TransformationProof [EQUIVALENT, 0 ms] (652) QDP (653) TransformationProof [EQUIVALENT, 0 ms] (654) QDP (655) TransformationProof [EQUIVALENT, 0 ms] (656) QDP (657) TransformationProof [EQUIVALENT, 1 ms] (658) QDP (659) TransformationProof [EQUIVALENT, 0 ms] (660) QDP (661) TransformationProof [EQUIVALENT, 0 ms] (662) QDP (663) TransformationProof [EQUIVALENT, 0 ms] (664) QDP (665) TransformationProof [EQUIVALENT, 0 ms] (666) QDP (667) TransformationProof [EQUIVALENT, 0 ms] (668) QDP (669) TransformationProof [EQUIVALENT, 0 ms] (670) QDP (671) TransformationProof [EQUIVALENT, 0 ms] (672) QDP (673) TransformationProof [EQUIVALENT, 0 ms] (674) QDP (675) TransformationProof [EQUIVALENT, 0 ms] (676) QDP (677) TransformationProof [EQUIVALENT, 0 ms] (678) QDP (679) TransformationProof [EQUIVALENT, 0 ms] (680) QDP (681) TransformationProof [EQUIVALENT, 0 ms] (682) QDP (683) TransformationProof [EQUIVALENT, 0 ms] (684) QDP (685) TransformationProof [EQUIVALENT, 0 ms] (686) QDP (687) TransformationProof [EQUIVALENT, 0 ms] (688) QDP (689) TransformationProof [EQUIVALENT, 0 ms] (690) QDP (691) TransformationProof [EQUIVALENT, 0 ms] (692) QDP (693) TransformationProof [EQUIVALENT, 0 ms] (694) QDP (695) TransformationProof [EQUIVALENT, 0 ms] (696) QDP (697) TransformationProof [EQUIVALENT, 0 ms] (698) QDP (699) TransformationProof [EQUIVALENT, 0 ms] (700) QDP (701) TransformationProof [EQUIVALENT, 0 ms] (702) QDP (703) TransformationProof [EQUIVALENT, 0 ms] (704) QDP (705) TransformationProof [EQUIVALENT, 0 ms] (706) QDP (707) TransformationProof [EQUIVALENT, 0 ms] (708) QDP (709) TransformationProof [EQUIVALENT, 0 ms] (710) QDP (711) TransformationProof [EQUIVALENT, 0 ms] (712) QDP (713) TransformationProof [EQUIVALENT, 0 ms] (714) QDP (715) TransformationProof [EQUIVALENT, 0 ms] (716) QDP (717) TransformationProof [EQUIVALENT, 3 ms] (718) QDP (719) TransformationProof [EQUIVALENT, 0 ms] (720) QDP (721) TransformationProof [EQUIVALENT, 0 ms] (722) QDP (723) TransformationProof [EQUIVALENT, 0 ms] (724) QDP (725) TransformationProof [EQUIVALENT, 0 ms] (726) QDP (727) TransformationProof [EQUIVALENT, 0 ms] (728) QDP (729) TransformationProof [EQUIVALENT, 0 ms] (730) QDP (731) TransformationProof [EQUIVALENT, 0 ms] (732) QDP (733) TransformationProof [EQUIVALENT, 0 ms] (734) QDP (735) TransformationProof [EQUIVALENT, 0 ms] (736) QDP (737) TransformationProof [EQUIVALENT, 0 ms] (738) QDP (739) TransformationProof [EQUIVALENT, 0 ms] (740) QDP (741) TransformationProof [EQUIVALENT, 0 ms] (742) QDP (743) TransformationProof [EQUIVALENT, 0 ms] (744) QDP (745) TransformationProof [EQUIVALENT, 0 ms] (746) QDP (747) TransformationProof [EQUIVALENT, 0 ms] (748) QDP (749) TransformationProof [EQUIVALENT, 0 ms] (750) QDP (751) TransformationProof [EQUIVALENT, 0 ms] (752) QDP (753) TransformationProof [EQUIVALENT, 0 ms] (754) QDP (755) TransformationProof [EQUIVALENT, 0 ms] (756) QDP (757) TransformationProof [EQUIVALENT, 0 ms] (758) QDP (759) TransformationProof [EQUIVALENT, 0 ms] (760) QDP (761) TransformationProof [EQUIVALENT, 0 ms] (762) QDP (763) TransformationProof [EQUIVALENT, 0 ms] (764) QDP (765) TransformationProof [EQUIVALENT, 0 ms] (766) QDP (767) TransformationProof [EQUIVALENT, 0 ms] (768) QDP (769) TransformationProof [EQUIVALENT, 0 ms] (770) QDP (771) TransformationProof [EQUIVALENT, 0 ms] (772) QDP (773) TransformationProof [EQUIVALENT, 0 ms] (774) QDP (775) TransformationProof [EQUIVALENT, 0 ms] (776) QDP (777) TransformationProof [EQUIVALENT, 0 ms] (778) QDP (779) TransformationProof [EQUIVALENT, 0 ms] (780) QDP (781) TransformationProof [EQUIVALENT, 0 ms] (782) QDP (783) TransformationProof [EQUIVALENT, 0 ms] (784) QDP (785) TransformationProof [EQUIVALENT, 0 ms] (786) QDP (787) TransformationProof [EQUIVALENT, 0 ms] (788) QDP (789) TransformationProof [EQUIVALENT, 0 ms] (790) QDP (791) TransformationProof [EQUIVALENT, 0 ms] (792) QDP (793) TransformationProof [EQUIVALENT, 0 ms] (794) QDP (795) TransformationProof [EQUIVALENT, 0 ms] (796) QDP (797) TransformationProof [EQUIVALENT, 0 ms] (798) QDP (799) TransformationProof [EQUIVALENT, 0 ms] (800) QDP (801) TransformationProof [EQUIVALENT, 0 ms] (802) QDP (803) TransformationProof [EQUIVALENT, 0 ms] (804) QDP (805) TransformationProof [EQUIVALENT, 0 ms] (806) QDP (807) TransformationProof [EQUIVALENT, 0 ms] (808) QDP (809) TransformationProof [EQUIVALENT, 0 ms] (810) QDP (811) TransformationProof [EQUIVALENT, 0 ms] (812) QDP (813) TransformationProof [EQUIVALENT, 0 ms] (814) QDP (815) TransformationProof [EQUIVALENT, 0 ms] (816) QDP (817) TransformationProof [EQUIVALENT, 0 ms] (818) QDP (819) TransformationProof [EQUIVALENT, 0 ms] (820) QDP (821) TransformationProof [EQUIVALENT, 0 ms] (822) QDP (823) TransformationProof [EQUIVALENT, 0 ms] (824) QDP (825) TransformationProof [EQUIVALENT, 0 ms] (826) QDP (827) TransformationProof [EQUIVALENT, 0 ms] (828) QDP (829) TransformationProof [EQUIVALENT, 0 ms] (830) QDP (831) TransformationProof [EQUIVALENT, 0 ms] (832) QDP (833) TransformationProof [EQUIVALENT, 0 ms] (834) QDP (835) TransformationProof [EQUIVALENT, 0 ms] (836) QDP (837) TransformationProof [EQUIVALENT, 0 ms] (838) QDP (839) TransformationProof [EQUIVALENT, 0 ms] (840) QDP (841) TransformationProof [EQUIVALENT, 1 ms] (842) QDP (843) TransformationProof [EQUIVALENT, 0 ms] (844) QDP (845) TransformationProof [EQUIVALENT, 0 ms] (846) QDP (847) TransformationProof [EQUIVALENT, 0 ms] (848) QDP (849) TransformationProof [EQUIVALENT, 0 ms] (850) QDP (851) TransformationProof [EQUIVALENT, 0 ms] (852) QDP (853) TransformationProof [EQUIVALENT, 0 ms] (854) QDP (855) TransformationProof [EQUIVALENT, 0 ms] (856) QDP (857) TransformationProof [EQUIVALENT, 0 ms] (858) QDP (859) TransformationProof [EQUIVALENT, 0 ms] (860) QDP (861) TransformationProof [EQUIVALENT, 0 ms] (862) QDP (863) TransformationProof [EQUIVALENT, 0 ms] (864) QDP (865) TransformationProof [EQUIVALENT, 0 ms] (866) QDP (867) TransformationProof [EQUIVALENT, 0 ms] (868) QDP (869) TransformationProof [EQUIVALENT, 0 ms] (870) QDP (871) TransformationProof [EQUIVALENT, 0 ms] (872) QDP (873) TransformationProof [EQUIVALENT, 0 ms] (874) QDP (875) TransformationProof [EQUIVALENT, 0 ms] (876) QDP (877) TransformationProof [EQUIVALENT, 0 ms] (878) QDP (879) TransformationProof [EQUIVALENT, 0 ms] (880) QDP (881) TransformationProof [EQUIVALENT, 0 ms] (882) QDP (883) TransformationProof [EQUIVALENT, 0 ms] (884) QDP (885) TransformationProof [EQUIVALENT, 0 ms] (886) QDP (887) TransformationProof [EQUIVALENT, 0 ms] (888) QDP (889) TransformationProof [EQUIVALENT, 0 ms] (890) QDP (891) TransformationProof [EQUIVALENT, 0 ms] (892) QDP (893) TransformationProof [EQUIVALENT, 0 ms] (894) QDP (895) TransformationProof [EQUIVALENT, 0 ms] (896) QDP (897) TransformationProof [EQUIVALENT, 3 ms] (898) QDP (899) TransformationProof [EQUIVALENT, 0 ms] (900) QDP (901) TransformationProof [EQUIVALENT, 0 ms] (902) QDP (903) TransformationProof [EQUIVALENT, 0 ms] (904) QDP (905) TransformationProof [EQUIVALENT, 0 ms] (906) QDP (907) TransformationProof [EQUIVALENT, 0 ms] (908) QDP (909) TransformationProof [EQUIVALENT, 0 ms] (910) QDP (911) TransformationProof [EQUIVALENT, 0 ms] (912) QDP (913) TransformationProof [EQUIVALENT, 0 ms] (914) QDP (915) TransformationProof [EQUIVALENT, 0 ms] (916) QDP (917) TransformationProof [EQUIVALENT, 0 ms] (918) QDP (919) TransformationProof [EQUIVALENT, 0 ms] (920) QDP (921) TransformationProof [EQUIVALENT, 0 ms] (922) QDP (923) TransformationProof [EQUIVALENT, 0 ms] (924) QDP (925) TransformationProof [EQUIVALENT, 0 ms] (926) QDP (927) TransformationProof [EQUIVALENT, 0 ms] (928) QDP (929) TransformationProof [EQUIVALENT, 0 ms] (930) QDP (931) TransformationProof [EQUIVALENT, 0 ms] (932) QDP (933) TransformationProof [EQUIVALENT, 0 ms] (934) QDP (935) TransformationProof [EQUIVALENT, 0 ms] (936) QDP (937) TransformationProof [EQUIVALENT, 0 ms] (938) QDP (939) TransformationProof [EQUIVALENT, 0 ms] (940) QDP (941) TransformationProof [EQUIVALENT, 0 ms] (942) QDP (943) TransformationProof [EQUIVALENT, 0 ms] (944) QDP (945) TransformationProof [EQUIVALENT, 0 ms] (946) QDP (947) TransformationProof [EQUIVALENT, 0 ms] (948) QDP (949) TransformationProof [EQUIVALENT, 0 ms] (950) QDP (951) TransformationProof [EQUIVALENT, 0 ms] (952) QDP (953) TransformationProof [EQUIVALENT, 0 ms] (954) QDP (955) TransformationProof [EQUIVALENT, 0 ms] (956) QDP (957) TransformationProof [EQUIVALENT, 0 ms] (958) QDP (959) TransformationProof [EQUIVALENT, 3 ms] (960) QDP (961) TransformationProof [EQUIVALENT, 0 ms] (962) QDP (963) TransformationProof [EQUIVALENT, 0 ms] (964) QDP (965) TransformationProof [EQUIVALENT, 0 ms] (966) QDP (967) TransformationProof [EQUIVALENT, 0 ms] (968) QDP (969) TransformationProof [EQUIVALENT, 0 ms] (970) QDP (971) TransformationProof [EQUIVALENT, 0 ms] (972) QDP (973) TransformationProof [EQUIVALENT, 0 ms] (974) QDP (975) TransformationProof [EQUIVALENT, 0 ms] (976) QDP (977) TransformationProof [EQUIVALENT, 0 ms] (978) QDP (979) TransformationProof [EQUIVALENT, 0 ms] (980) QDP (981) TransformationProof [EQUIVALENT, 0 ms] (982) QDP (983) TransformationProof [EQUIVALENT, 0 ms] (984) QDP (985) TransformationProof [EQUIVALENT, 0 ms] (986) QDP (987) TransformationProof [EQUIVALENT, 0 ms] (988) QDP (989) TransformationProof [EQUIVALENT, 0 ms] (990) QDP (991) TransformationProof [EQUIVALENT, 0 ms] (992) QDP (993) TransformationProof [EQUIVALENT, 0 ms] (994) QDP (995) TransformationProof [EQUIVALENT, 0 ms] (996) QDP (997) TransformationProof [EQUIVALENT, 0 ms] (998) QDP (999) TransformationProof [EQUIVALENT, 0 ms] (1000) QDP (1001) TransformationProof [EQUIVALENT, 0 ms] (1002) QDP (1003) TransformationProof [EQUIVALENT, 0 ms] (1004) QDP (1005) TransformationProof [EQUIVALENT, 0 ms] (1006) QDP (1007) TransformationProof [EQUIVALENT, 0 ms] (1008) QDP (1009) TransformationProof [EQUIVALENT, 0 ms] (1010) QDP (1011) TransformationProof [EQUIVALENT, 0 ms] (1012) QDP (1013) TransformationProof [EQUIVALENT, 0 ms] (1014) QDP (1015) TransformationProof [EQUIVALENT, 0 ms] (1016) QDP (1017) TransformationProof [EQUIVALENT, 0 ms] (1018) QDP (1019) TransformationProof [EQUIVALENT, 0 ms] (1020) QDP (1021) TransformationProof [EQUIVALENT, 1 ms] (1022) QDP (1023) TransformationProof [EQUIVALENT, 0 ms] (1024) QDP (1025) TransformationProof [EQUIVALENT, 0 ms] (1026) QDP (1027) TransformationProof [EQUIVALENT, 0 ms] (1028) QDP (1029) TransformationProof [EQUIVALENT, 0 ms] (1030) QDP (1031) TransformationProof [EQUIVALENT, 0 ms] (1032) QDP (1033) TransformationProof [EQUIVALENT, 0 ms] (1034) QDP (1035) TransformationProof [EQUIVALENT, 0 ms] (1036) QDP (1037) TransformationProof [EQUIVALENT, 0 ms] (1038) QDP (1039) TransformationProof [EQUIVALENT, 0 ms] (1040) QDP (1041) TransformationProof [EQUIVALENT, 0 ms] (1042) QDP (1043) TransformationProof [EQUIVALENT, 0 ms] (1044) QDP (1045) TransformationProof [EQUIVALENT, 0 ms] (1046) QDP (1047) TransformationProof [EQUIVALENT, 0 ms] (1048) QDP (1049) TransformationProof [EQUIVALENT, 0 ms] (1050) QDP (1051) TransformationProof [EQUIVALENT, 0 ms] (1052) QDP (1053) TransformationProof [EQUIVALENT, 0 ms] (1054) QDP (1055) TransformationProof [EQUIVALENT, 0 ms] (1056) QDP (1057) TransformationProof [EQUIVALENT, 0 ms] (1058) QDP (1059) TransformationProof [EQUIVALENT, 0 ms] (1060) QDP (1061) TransformationProof [EQUIVALENT, 0 ms] (1062) QDP (1063) TransformationProof [EQUIVALENT, 0 ms] (1064) QDP (1065) TransformationProof [EQUIVALENT, 0 ms] (1066) QDP (1067) TransformationProof [EQUIVALENT, 0 ms] (1068) QDP (1069) TransformationProof [EQUIVALENT, 0 ms] (1070) QDP (1071) TransformationProof [EQUIVALENT, 0 ms] (1072) QDP (1073) TransformationProof [EQUIVALENT, 0 ms] (1074) QDP (1075) TransformationProof [EQUIVALENT, 0 ms] (1076) QDP (1077) TransformationProof [EQUIVALENT, 1 ms] (1078) QDP (1079) TransformationProof [EQUIVALENT, 0 ms] (1080) QDP (1081) TransformationProof [EQUIVALENT, 0 ms] (1082) QDP (1083) TransformationProof [EQUIVALENT, 0 ms] (1084) QDP (1085) TransformationProof [EQUIVALENT, 0 ms] (1086) QDP (1087) TransformationProof [EQUIVALENT, 0 ms] (1088) QDP (1089) TransformationProof [EQUIVALENT, 0 ms] (1090) QDP (1091) TransformationProof [EQUIVALENT, 0 ms] (1092) QDP (1093) TransformationProof [EQUIVALENT, 0 ms] (1094) QDP (1095) TransformationProof [EQUIVALENT, 0 ms] (1096) QDP (1097) TransformationProof [EQUIVALENT, 0 ms] (1098) QDP (1099) TransformationProof [EQUIVALENT, 0 ms] (1100) QDP (1101) TransformationProof [EQUIVALENT, 0 ms] (1102) QDP (1103) TransformationProof [EQUIVALENT, 0 ms] (1104) QDP (1105) TransformationProof [EQUIVALENT, 0 ms] (1106) QDP (1107) TransformationProof [EQUIVALENT, 0 ms] (1108) QDP (1109) TransformationProof [EQUIVALENT, 0 ms] (1110) QDP (1111) TransformationProof [EQUIVALENT, 0 ms] (1112) QDP (1113) TransformationProof [EQUIVALENT, 0 ms] (1114) QDP (1115) TransformationProof [EQUIVALENT, 0 ms] (1116) QDP (1117) TransformationProof [EQUIVALENT, 0 ms] (1118) QDP (1119) TransformationProof [EQUIVALENT, 0 ms] (1120) QDP (1121) TransformationProof [EQUIVALENT, 0 ms] (1122) QDP (1123) TransformationProof [EQUIVALENT, 0 ms] (1124) QDP (1125) TransformationProof [EQUIVALENT, 0 ms] (1126) QDP (1127) TransformationProof [EQUIVALENT, 0 ms] (1128) QDP (1129) TransformationProof [EQUIVALENT, 0 ms] (1130) QDP (1131) TransformationProof [EQUIVALENT, 0 ms] (1132) QDP (1133) TransformationProof [EQUIVALENT, 3 ms] (1134) QDP (1135) TransformationProof [EQUIVALENT, 0 ms] (1136) QDP (1137) TransformationProof [EQUIVALENT, 0 ms] (1138) QDP (1139) TransformationProof [EQUIVALENT, 0 ms] (1140) QDP (1141) TransformationProof [EQUIVALENT, 0 ms] (1142) QDP (1143) TransformationProof [EQUIVALENT, 0 ms] (1144) QDP (1145) TransformationProof [EQUIVALENT, 0 ms] (1146) QDP (1147) TransformationProof [EQUIVALENT, 0 ms] (1148) QDP (1149) TransformationProof [EQUIVALENT, 0 ms] (1150) QDP (1151) TransformationProof [EQUIVALENT, 0 ms] (1152) QDP (1153) TransformationProof [EQUIVALENT, 0 ms] (1154) QDP (1155) TransformationProof [EQUIVALENT, 0 ms] (1156) QDP (1157) TransformationProof [EQUIVALENT, 0 ms] (1158) QDP (1159) TransformationProof [EQUIVALENT, 0 ms] (1160) QDP (1161) TransformationProof [EQUIVALENT, 0 ms] (1162) QDP (1163) TransformationProof [EQUIVALENT, 0 ms] (1164) QDP (1165) TransformationProof [EQUIVALENT, 0 ms] (1166) QDP (1167) TransformationProof [EQUIVALENT, 0 ms] (1168) QDP (1169) TransformationProof [EQUIVALENT, 0 ms] (1170) QDP (1171) TransformationProof [EQUIVALENT, 0 ms] (1172) QDP (1173) TransformationProof [EQUIVALENT, 0 ms] (1174) QDP (1175) TransformationProof [EQUIVALENT, 0 ms] (1176) QDP (1177) TransformationProof [EQUIVALENT, 0 ms] (1178) QDP (1179) TransformationProof [EQUIVALENT, 0 ms] (1180) QDP (1181) DependencyGraphProof [EQUIVALENT, 0 ms] (1182) QDP (1183) TransformationProof [EQUIVALENT, 0 ms] (1184) QDP (1185) TransformationProof [EQUIVALENT, 0 ms] (1186) QDP (1187) TransformationProof [EQUIVALENT, 0 ms] (1188) QDP (1189) TransformationProof [EQUIVALENT, 0 ms] (1190) QDP (1191) TransformationProof [EQUIVALENT, 0 ms] (1192) QDP (1193) TransformationProof [EQUIVALENT, 1 ms] (1194) QDP (1195) TransformationProof [EQUIVALENT, 0 ms] (1196) QDP (1197) TransformationProof [EQUIVALENT, 0 ms] (1198) QDP (1199) TransformationProof [EQUIVALENT, 0 ms] (1200) QDP (1201) TransformationProof [EQUIVALENT, 0 ms] (1202) QDP (1203) TransformationProof [EQUIVALENT, 0 ms] (1204) QDP (1205) TransformationProof [EQUIVALENT, 0 ms] (1206) QDP (1207) TransformationProof [EQUIVALENT, 0 ms] (1208) QDP (1209) TransformationProof [EQUIVALENT, 0 ms] (1210) QDP (1211) TransformationProof [EQUIVALENT, 0 ms] (1212) QDP (1213) TransformationProof [EQUIVALENT, 0 ms] (1214) QDP (1215) TransformationProof [EQUIVALENT, 0 ms] (1216) QDP (1217) TransformationProof [EQUIVALENT, 0 ms] (1218) QDP (1219) TransformationProof [EQUIVALENT, 0 ms] (1220) QDP (1221) TransformationProof [EQUIVALENT, 0 ms] (1222) QDP (1223) TransformationProof [EQUIVALENT, 0 ms] (1224) QDP (1225) TransformationProof [EQUIVALENT, 0 ms] (1226) QDP (1227) TransformationProof [EQUIVALENT, 0 ms] (1228) QDP (1229) DependencyGraphProof [EQUIVALENT, 0 ms] (1230) QDP (1231) TransformationProof [EQUIVALENT, 0 ms] (1232) QDP (1233) TransformationProof [EQUIVALENT, 0 ms] (1234) QDP (1235) TransformationProof [EQUIVALENT, 0 ms] (1236) QDP (1237) TransformationProof [EQUIVALENT, 0 ms] (1238) QDP (1239) TransformationProof [EQUIVALENT, 0 ms] (1240) QDP (1241) TransformationProof [EQUIVALENT, 0 ms] (1242) QDP (1243) TransformationProof [EQUIVALENT, 0 ms] (1244) QDP (1245) TransformationProof [EQUIVALENT, 0 ms] (1246) QDP (1247) TransformationProof [EQUIVALENT, 0 ms] (1248) QDP (1249) TransformationProof [EQUIVALENT, 0 ms] (1250) QDP (1251) TransformationProof [EQUIVALENT, 0 ms] (1252) QDP (1253) TransformationProof [EQUIVALENT, 0 ms] (1254) QDP (1255) TransformationProof [EQUIVALENT, 0 ms] (1256) QDP (1257) TransformationProof [EQUIVALENT, 0 ms] (1258) QDP (1259) TransformationProof [EQUIVALENT, 0 ms] (1260) QDP (1261) TransformationProof [EQUIVALENT, 0 ms] (1262) QDP (1263) TransformationProof [EQUIVALENT, 0 ms] (1264) QDP (1265) TransformationProof [EQUIVALENT, 0 ms] (1266) QDP (1267) TransformationProof [EQUIVALENT, 0 ms] (1268) QDP (1269) TransformationProof [EQUIVALENT, 0 ms] (1270) QDP (1271) TransformationProof [EQUIVALENT, 0 ms] (1272) QDP (1273) TransformationProof [EQUIVALENT, 0 ms] (1274) QDP (1275) TransformationProof [EQUIVALENT, 0 ms] (1276) QDP (1277) TransformationProof [EQUIVALENT, 0 ms] (1278) QDP (1279) TransformationProof [EQUIVALENT, 0 ms] (1280) QDP (1281) TransformationProof [EQUIVALENT, 0 ms] (1282) QDP (1283) TransformationProof [EQUIVALENT, 0 ms] (1284) QDP (1285) TransformationProof [EQUIVALENT, 0 ms] (1286) QDP (1287) TransformationProof [EQUIVALENT, 0 ms] (1288) QDP (1289) DependencyGraphProof [EQUIVALENT, 0 ms] (1290) QDP (1291) TransformationProof [EQUIVALENT, 0 ms] (1292) QDP (1293) TransformationProof [EQUIVALENT, 0 ms] (1294) QDP (1295) TransformationProof [EQUIVALENT, 0 ms] (1296) QDP (1297) TransformationProof [EQUIVALENT, 0 ms] (1298) QDP (1299) TransformationProof [EQUIVALENT, 0 ms] (1300) QDP (1301) TransformationProof [EQUIVALENT, 0 ms] (1302) QDP (1303) TransformationProof [EQUIVALENT, 0 ms] (1304) QDP (1305) TransformationProof [EQUIVALENT, 0 ms] (1306) QDP (1307) TransformationProof [EQUIVALENT, 0 ms] (1308) QDP (1309) TransformationProof [EQUIVALENT, 0 ms] (1310) QDP (1311) TransformationProof [EQUIVALENT, 0 ms] (1312) QDP (1313) TransformationProof [EQUIVALENT, 0 ms] (1314) QDP (1315) TransformationProof [EQUIVALENT, 0 ms] (1316) QDP (1317) TransformationProof [EQUIVALENT, 0 ms] (1318) QDP (1319) TransformationProof [EQUIVALENT, 0 ms] (1320) QDP (1321) TransformationProof [EQUIVALENT, 0 ms] (1322) QDP (1323) TransformationProof [EQUIVALENT, 0 ms] (1324) QDP (1325) TransformationProof [EQUIVALENT, 0 ms] (1326) QDP (1327) TransformationProof [EQUIVALENT, 0 ms] (1328) QDP (1329) TransformationProof [EQUIVALENT, 0 ms] (1330) QDP (1331) TransformationProof [EQUIVALENT, 0 ms] (1332) QDP (1333) TransformationProof [EQUIVALENT, 0 ms] (1334) QDP (1335) TransformationProof [EQUIVALENT, 0 ms] (1336) QDP (1337) TransformationProof [EQUIVALENT, 0 ms] (1338) QDP (1339) TransformationProof [EQUIVALENT, 0 ms] (1340) QDP (1341) TransformationProof [EQUIVALENT, 0 ms] (1342) QDP (1343) TransformationProof [EQUIVALENT, 0 ms] (1344) QDP (1345) TransformationProof [EQUIVALENT, 0 ms] (1346) QDP (1347) TransformationProof [EQUIVALENT, 0 ms] (1348) QDP (1349) TransformationProof [EQUIVALENT, 0 ms] (1350) QDP (1351) TransformationProof [EQUIVALENT, 0 ms] (1352) QDP (1353) TransformationProof [EQUIVALENT, 0 ms] (1354) QDP (1355) TransformationProof [EQUIVALENT, 0 ms] (1356) QDP (1357) TransformationProof [EQUIVALENT, 0 ms] (1358) QDP (1359) TransformationProof [EQUIVALENT, 0 ms] (1360) QDP (1361) TransformationProof [EQUIVALENT, 0 ms] (1362) QDP (1363) TransformationProof [EQUIVALENT, 0 ms] (1364) QDP (1365) TransformationProof [EQUIVALENT, 0 ms] (1366) QDP (1367) TransformationProof [EQUIVALENT, 0 ms] (1368) QDP (1369) TransformationProof [EQUIVALENT, 0 ms] (1370) QDP (1371) TransformationProof [EQUIVALENT, 0 ms] (1372) QDP (1373) TransformationProof [EQUIVALENT, 0 ms] (1374) QDP (1375) TransformationProof [EQUIVALENT, 0 ms] (1376) QDP (1377) TransformationProof [EQUIVALENT, 0 ms] (1378) QDP (1379) TransformationProof [EQUIVALENT, 0 ms] (1380) QDP (1381) TransformationProof [EQUIVALENT, 0 ms] (1382) QDP (1383) TransformationProof [EQUIVALENT, 0 ms] (1384) QDP (1385) TransformationProof [EQUIVALENT, 0 ms] (1386) QDP (1387) TransformationProof [EQUIVALENT, 0 ms] (1388) QDP (1389) TransformationProof [EQUIVALENT, 0 ms] (1390) QDP (1391) TransformationProof [EQUIVALENT, 0 ms] (1392) QDP (1393) TransformationProof [EQUIVALENT, 0 ms] (1394) QDP (1395) TransformationProof [EQUIVALENT, 0 ms] (1396) QDP (1397) TransformationProof [EQUIVALENT, 0 ms] (1398) QDP (1399) TransformationProof [EQUIVALENT, 0 ms] (1400) QDP (1401) TransformationProof [EQUIVALENT, 0 ms] (1402) QDP (1403) TransformationProof [EQUIVALENT, 0 ms] (1404) QDP (1405) TransformationProof [EQUIVALENT, 0 ms] (1406) QDP (1407) TransformationProof [EQUIVALENT, 0 ms] (1408) QDP (1409) TransformationProof [EQUIVALENT, 1 ms] (1410) QDP (1411) TransformationProof [EQUIVALENT, 0 ms] (1412) QDP (1413) TransformationProof [EQUIVALENT, 0 ms] (1414) QDP (1415) TransformationProof [EQUIVALENT, 0 ms] (1416) QDP (1417) TransformationProof [EQUIVALENT, 0 ms] (1418) QDP (1419) TransformationProof [EQUIVALENT, 0 ms] (1420) QDP (1421) TransformationProof [EQUIVALENT, 0 ms] (1422) QDP (1423) TransformationProof [EQUIVALENT, 0 ms] (1424) QDP (1425) TransformationProof [EQUIVALENT, 0 ms] (1426) QDP (1427) TransformationProof [EQUIVALENT, 0 ms] (1428) QDP (1429) TransformationProof [EQUIVALENT, 0 ms] (1430) QDP (1431) TransformationProof [EQUIVALENT, 0 ms] (1432) QDP (1433) TransformationProof [EQUIVALENT, 0 ms] (1434) QDP (1435) TransformationProof [EQUIVALENT, 0 ms] (1436) QDP (1437) TransformationProof [EQUIVALENT, 0 ms] (1438) QDP (1439) TransformationProof [EQUIVALENT, 0 ms] (1440) QDP (1441) TransformationProof [EQUIVALENT, 0 ms] (1442) QDP (1443) TransformationProof [EQUIVALENT, 0 ms] (1444) QDP (1445) TransformationProof [EQUIVALENT, 0 ms] (1446) QDP (1447) TransformationProof [EQUIVALENT, 0 ms] (1448) QDP (1449) TransformationProof [EQUIVALENT, 0 ms] (1450) QDP (1451) TransformationProof [EQUIVALENT, 0 ms] (1452) QDP (1453) TransformationProof [EQUIVALENT, 0 ms] (1454) QDP (1455) TransformationProof [EQUIVALENT, 0 ms] (1456) QDP (1457) TransformationProof [EQUIVALENT, 0 ms] (1458) QDP (1459) TransformationProof [EQUIVALENT, 0 ms] (1460) QDP (1461) TransformationProof [EQUIVALENT, 0 ms] (1462) QDP (1463) DependencyGraphProof [EQUIVALENT, 0 ms] (1464) QDP (1465) TransformationProof [EQUIVALENT, 0 ms] (1466) QDP (1467) TransformationProof [EQUIVALENT, 0 ms] (1468) QDP (1469) TransformationProof [EQUIVALENT, 0 ms] (1470) QDP (1471) TransformationProof [EQUIVALENT, 0 ms] (1472) QDP (1473) TransformationProof [EQUIVALENT, 0 ms] (1474) QDP (1475) TransformationProof [EQUIVALENT, 0 ms] (1476) QDP (1477) TransformationProof [EQUIVALENT, 0 ms] (1478) QDP (1479) TransformationProof [EQUIVALENT, 0 ms] (1480) QDP (1481) TransformationProof [EQUIVALENT, 1 ms] (1482) QDP (1483) TransformationProof [EQUIVALENT, 0 ms] (1484) QDP (1485) TransformationProof [EQUIVALENT, 0 ms] (1486) QDP (1487) DependencyGraphProof [EQUIVALENT, 0 ms] (1488) QDP (1489) TransformationProof [EQUIVALENT, 0 ms] (1490) QDP (1491) DependencyGraphProof [EQUIVALENT, 0 ms] (1492) QDP (1493) TransformationProof [EQUIVALENT, 0 ms] (1494) QDP (1495) TransformationProof [EQUIVALENT, 0 ms] (1496) QDP (1497) TransformationProof [EQUIVALENT, 0 ms] (1498) QDP (1499) TransformationProof [EQUIVALENT, 0 ms] (1500) QDP (1501) TransformationProof [EQUIVALENT, 0 ms] (1502) QDP (1503) TransformationProof [EQUIVALENT, 0 ms] (1504) QDP (1505) TransformationProof [EQUIVALENT, 0 ms] (1506) QDP (1507) TransformationProof [EQUIVALENT, 0 ms] (1508) QDP (1509) TransformationProof [EQUIVALENT, 0 ms] (1510) QDP (1511) TransformationProof [EQUIVALENT, 0 ms] (1512) QDP (1513) TransformationProof [EQUIVALENT, 0 ms] (1514) QDP (1515) TransformationProof [EQUIVALENT, 0 ms] (1516) QDP (1517) TransformationProof [EQUIVALENT, 0 ms] (1518) QDP (1519) DependencyGraphProof [EQUIVALENT, 0 ms] (1520) QDP (1521) TransformationProof [EQUIVALENT, 0 ms] (1522) QDP (1523) TransformationProof [EQUIVALENT, 0 ms] (1524) QDP (1525) TransformationProof [EQUIVALENT, 0 ms] (1526) QDP (1527) TransformationProof [EQUIVALENT, 0 ms] (1528) QDP (1529) TransformationProof [EQUIVALENT, 0 ms] (1530) QDP (1531) TransformationProof [EQUIVALENT, 0 ms] (1532) QDP (1533) TransformationProof [EQUIVALENT, 0 ms] (1534) QDP (1535) TransformationProof [EQUIVALENT, 0 ms] (1536) QDP (1537) TransformationProof [EQUIVALENT, 0 ms] (1538) QDP (1539) TransformationProof [EQUIVALENT, 0 ms] (1540) QDP (1541) TransformationProof [EQUIVALENT, 0 ms] (1542) QDP (1543) TransformationProof [EQUIVALENT, 0 ms] (1544) QDP (1545) TransformationProof [EQUIVALENT, 2 ms] (1546) QDP (1547) TransformationProof [EQUIVALENT, 0 ms] (1548) QDP (1549) TransformationProof [EQUIVALENT, 0 ms] (1550) QDP (1551) TransformationProof [EQUIVALENT, 0 ms] (1552) QDP (1553) TransformationProof [EQUIVALENT, 0 ms] (1554) QDP (1555) TransformationProof [EQUIVALENT, 0 ms] (1556) QDP (1557) TransformationProof [EQUIVALENT, 0 ms] (1558) QDP (1559) DependencyGraphProof [EQUIVALENT, 0 ms] (1560) QDP (1561) TransformationProof [EQUIVALENT, 0 ms] (1562) QDP (1563) TransformationProof [EQUIVALENT, 0 ms] (1564) QDP (1565) DependencyGraphProof [EQUIVALENT, 0 ms] (1566) QDP (1567) TransformationProof [EQUIVALENT, 0 ms] (1568) QDP (1569) DependencyGraphProof [EQUIVALENT, 0 ms] (1570) QDP (1571) TransformationProof [EQUIVALENT, 0 ms] (1572) QDP (1573) TransformationProof [EQUIVALENT, 0 ms] (1574) QDP (1575) TransformationProof [EQUIVALENT, 0 ms] (1576) QDP (1577) DependencyGraphProof [EQUIVALENT, 0 ms] (1578) QDP (1579) TransformationProof [EQUIVALENT, 0 ms] (1580) QDP (1581) TransformationProof [EQUIVALENT, 0 ms] (1582) QDP (1583) DependencyGraphProof [EQUIVALENT, 0 ms] (1584) QDP (1585) TransformationProof [EQUIVALENT, 0 ms] (1586) QDP (1587) DependencyGraphProof [EQUIVALENT, 0 ms] (1588) QDP (1589) TransformationProof [EQUIVALENT, 0 ms] (1590) QDP (1591) TransformationProof [EQUIVALENT, 0 ms] (1592) QDP (1593) DependencyGraphProof [EQUIVALENT, 0 ms] (1594) QDP (1595) TransformationProof [EQUIVALENT, 0 ms] (1596) QDP (1597) TransformationProof [EQUIVALENT, 0 ms] (1598) QDP (1599) TransformationProof [EQUIVALENT, 0 ms] (1600) QDP (1601) TransformationProof [EQUIVALENT, 0 ms] (1602) QDP (1603) TransformationProof [EQUIVALENT, 0 ms] (1604) QDP (1605) TransformationProof [EQUIVALENT, 0 ms] (1606) QDP (1607) DependencyGraphProof [EQUIVALENT, 0 ms] (1608) QDP (1609) TransformationProof [EQUIVALENT, 0 ms] (1610) QDP (1611) TransformationProof [EQUIVALENT, 1 ms] (1612) QDP (1613) TransformationProof [EQUIVALENT, 0 ms] (1614) QDP (1615) DependencyGraphProof [EQUIVALENT, 0 ms] (1616) QDP (1617) TransformationProof [EQUIVALENT, 0 ms] (1618) QDP (1619) DependencyGraphProof [EQUIVALENT, 0 ms] (1620) QDP (1621) TransformationProof [EQUIVALENT, 0 ms] (1622) QDP (1623) TransformationProof [EQUIVALENT, 0 ms] (1624) QDP (1625) TransformationProof [EQUIVALENT, 0 ms] (1626) QDP (1627) TransformationProof [EQUIVALENT, 0 ms] (1628) QDP (1629) TransformationProof [EQUIVALENT, 0 ms] (1630) QDP (1631) TransformationProof [EQUIVALENT, 0 ms] (1632) QDP (1633) TransformationProof [EQUIVALENT, 0 ms] (1634) QDP (1635) TransformationProof [EQUIVALENT, 0 ms] (1636) QDP (1637) TransformationProof [EQUIVALENT, 0 ms] (1638) QDP (1639) TransformationProof [EQUIVALENT, 0 ms] (1640) QDP (1641) TransformationProof [EQUIVALENT, 0 ms] (1642) QDP (1643) TransformationProof [EQUIVALENT, 0 ms] (1644) QDP (1645) TransformationProof [EQUIVALENT, 0 ms] (1646) QDP (1647) TransformationProof [EQUIVALENT, 0 ms] (1648) QDP (1649) TransformationProof [EQUIVALENT, 0 ms] (1650) QDP (1651) TransformationProof [EQUIVALENT, 0 ms] (1652) QDP (1653) DependencyGraphProof [EQUIVALENT, 0 ms] (1654) QDP (1655) TransformationProof [EQUIVALENT, 0 ms] (1656) QDP (1657) DependencyGraphProof [EQUIVALENT, 0 ms] (1658) QDP (1659) TransformationProof [EQUIVALENT, 0 ms] (1660) QDP (1661) TransformationProof [EQUIVALENT, 0 ms] (1662) QDP (1663) DependencyGraphProof [EQUIVALENT, 0 ms] (1664) QDP (1665) TransformationProof [EQUIVALENT, 0 ms] (1666) QDP (1667) TransformationProof [EQUIVALENT, 0 ms] (1668) QDP (1669) TransformationProof [EQUIVALENT, 0 ms] (1670) QDP (1671) TransformationProof [EQUIVALENT, 0 ms] (1672) QDP (1673) TransformationProof [EQUIVALENT, 0 ms] (1674) QDP (1675) TransformationProof [EQUIVALENT, 0 ms] (1676) QDP (1677) TransformationProof [EQUIVALENT, 0 ms] (1678) QDP (1679) TransformationProof [EQUIVALENT, 0 ms] (1680) QDP (1681) DependencyGraphProof [EQUIVALENT, 0 ms] (1682) QDP (1683) TransformationProof [EQUIVALENT, 0 ms] (1684) QDP (1685) DependencyGraphProof [EQUIVALENT, 0 ms] (1686) QDP (1687) TransformationProof [EQUIVALENT, 0 ms] (1688) QDP (1689) TransformationProof [EQUIVALENT, 0 ms] (1690) QDP (1691) TransformationProof [EQUIVALENT, 0 ms] (1692) QDP (1693) TransformationProof [EQUIVALENT, 0 ms] (1694) QDP (1695) TransformationProof [EQUIVALENT, 0 ms] (1696) QDP (1697) TransformationProof [EQUIVALENT, 0 ms] (1698) QDP (1699) TransformationProof [EQUIVALENT, 0 ms] (1700) QDP (1701) TransformationProof [EQUIVALENT, 0 ms] (1702) QDP (1703) TransformationProof [EQUIVALENT, 0 ms] (1704) QDP (1705) TransformationProof [EQUIVALENT, 0 ms] (1706) QDP (1707) TransformationProof [EQUIVALENT, 0 ms] (1708) QDP (1709) TransformationProof [EQUIVALENT, 0 ms] (1710) QDP (1711) TransformationProof [EQUIVALENT, 0 ms] (1712) QDP (1713) TransformationProof [EQUIVALENT, 0 ms] (1714) QDP (1715) TransformationProof [EQUIVALENT, 0 ms] (1716) QDP (1717) TransformationProof [EQUIVALENT, 0 ms] (1718) QDP (1719) TransformationProof [EQUIVALENT, 0 ms] (1720) QDP (1721) TransformationProof [EQUIVALENT, 0 ms] (1722) QDP (1723) TransformationProof [EQUIVALENT, 0 ms] (1724) QDP (1725) TransformationProof [EQUIVALENT, 0 ms] (1726) QDP (1727) DependencyGraphProof [EQUIVALENT, 0 ms] (1728) QDP (1729) TransformationProof [EQUIVALENT, 0 ms] (1730) QDP (1731) TransformationProof [EQUIVALENT, 0 ms] (1732) QDP (1733) TransformationProof [EQUIVALENT, 0 ms] (1734) QDP (1735) TransformationProof [EQUIVALENT, 0 ms] (1736) QDP (1737) TransformationProof [EQUIVALENT, 0 ms] (1738) QDP (1739) TransformationProof [EQUIVALENT, 0 ms] (1740) QDP (1741) TransformationProof [EQUIVALENT, 0 ms] (1742) QDP (1743) TransformationProof [EQUIVALENT, 0 ms] (1744) QDP (1745) TransformationProof [EQUIVALENT, 0 ms] (1746) QDP (1747) TransformationProof [EQUIVALENT, 0 ms] (1748) QDP (1749) TransformationProof [EQUIVALENT, 0 ms] (1750) QDP (1751) TransformationProof [EQUIVALENT, 0 ms] (1752) QDP (1753) TransformationProof [EQUIVALENT, 0 ms] (1754) QDP (1755) TransformationProof [EQUIVALENT, 0 ms] (1756) QDP (1757) TransformationProof [EQUIVALENT, 0 ms] (1758) QDP (1759) TransformationProof [EQUIVALENT, 0 ms] (1760) QDP (1761) TransformationProof [EQUIVALENT, 0 ms] (1762) QDP (1763) TransformationProof [EQUIVALENT, 0 ms] (1764) QDP (1765) TransformationProof [EQUIVALENT, 0 ms] (1766) QDP (1767) TransformationProof [EQUIVALENT, 0 ms] (1768) QDP (1769) TransformationProof [EQUIVALENT, 0 ms] (1770) QDP (1771) TransformationProof [EQUIVALENT, 0 ms] (1772) QDP (1773) DependencyGraphProof [EQUIVALENT, 1 ms] (1774) QDP (1775) TransformationProof [EQUIVALENT, 0 ms] (1776) QDP (1777) TransformationProof [EQUIVALENT, 0 ms] (1778) QDP (1779) TransformationProof [EQUIVALENT, 0 ms] (1780) QDP (1781) TransformationProof [EQUIVALENT, 0 ms] (1782) QDP (1783) TransformationProof [EQUIVALENT, 0 ms] (1784) QDP (1785) TransformationProof [EQUIVALENT, 0 ms] (1786) QDP (1787) TransformationProof [EQUIVALENT, 0 ms] (1788) QDP (1789) TransformationProof [EQUIVALENT, 0 ms] (1790) QDP (1791) TransformationProof [EQUIVALENT, 0 ms] (1792) QDP (1793) TransformationProof [EQUIVALENT, 0 ms] (1794) QDP (1795) TransformationProof [EQUIVALENT, 0 ms] (1796) QDP (1797) TransformationProof [EQUIVALENT, 0 ms] (1798) QDP (1799) TransformationProof [EQUIVALENT, 0 ms] (1800) QDP (1801) TransformationProof [EQUIVALENT, 0 ms] (1802) QDP (1803) TransformationProof [EQUIVALENT, 0 ms] (1804) QDP (1805) TransformationProof [EQUIVALENT, 0 ms] (1806) QDP (1807) TransformationProof [EQUIVALENT, 0 ms] (1808) QDP (1809) TransformationProof [EQUIVALENT, 0 ms] (1810) QDP (1811) TransformationProof [EQUIVALENT, 0 ms] (1812) QDP (1813) TransformationProof [EQUIVALENT, 0 ms] (1814) QDP (1815) TransformationProof [EQUIVALENT, 0 ms] (1816) QDP (1817) TransformationProof [EQUIVALENT, 0 ms] (1818) QDP (1819) TransformationProof [EQUIVALENT, 0 ms] (1820) QDP (1821) TransformationProof [EQUIVALENT, 0 ms] (1822) QDP (1823) TransformationProof [EQUIVALENT, 0 ms] (1824) QDP (1825) TransformationProof [EQUIVALENT, 0 ms] (1826) QDP (1827) TransformationProof [EQUIVALENT, 0 ms] (1828) QDP (1829) TransformationProof [EQUIVALENT, 0 ms] (1830) QDP (1831) TransformationProof [EQUIVALENT, 0 ms] (1832) QDP (1833) TransformationProof [EQUIVALENT, 0 ms] (1834) QDP (1835) TransformationProof [EQUIVALENT, 0 ms] (1836) QDP (1837) TransformationProof [EQUIVALENT, 0 ms] (1838) QDP (1839) TransformationProof [EQUIVALENT, 0 ms] (1840) QDP (1841) TransformationProof [EQUIVALENT, 0 ms] (1842) QDP (1843) TransformationProof [EQUIVALENT, 0 ms] (1844) QDP (1845) TransformationProof [EQUIVALENT, 0 ms] (1846) QDP (1847) TransformationProof [EQUIVALENT, 0 ms] (1848) QDP (1849) TransformationProof [EQUIVALENT, 0 ms] (1850) QDP (1851) TransformationProof [EQUIVALENT, 0 ms] (1852) QDP (1853) TransformationProof [EQUIVALENT, 0 ms] (1854) QDP (1855) TransformationProof [EQUIVALENT, 0 ms] (1856) QDP (1857) TransformationProof [EQUIVALENT, 0 ms] (1858) QDP (1859) TransformationProof [EQUIVALENT, 0 ms] (1860) QDP (1861) TransformationProof [EQUIVALENT, 0 ms] (1862) QDP (1863) TransformationProof [EQUIVALENT, 0 ms] (1864) QDP (1865) TransformationProof [EQUIVALENT, 0 ms] (1866) QDP (1867) TransformationProof [EQUIVALENT, 0 ms] (1868) QDP (1869) TransformationProof [EQUIVALENT, 2 ms] (1870) QDP (1871) TransformationProof [EQUIVALENT, 0 ms] (1872) QDP (1873) TransformationProof [EQUIVALENT, 0 ms] (1874) QDP (1875) TransformationProof [EQUIVALENT, 0 ms] (1876) QDP (1877) TransformationProof [EQUIVALENT, 0 ms] (1878) QDP (1879) TransformationProof [EQUIVALENT, 0 ms] (1880) QDP (1881) TransformationProof [EQUIVALENT, 0 ms] (1882) QDP (1883) TransformationProof [EQUIVALENT, 0 ms] (1884) QDP (1885) TransformationProof [EQUIVALENT, 0 ms] (1886) QDP (1887) TransformationProof [EQUIVALENT, 0 ms] (1888) QDP (1889) TransformationProof [EQUIVALENT, 0 ms] (1890) QDP (1891) DependencyGraphProof [EQUIVALENT, 0 ms] (1892) QDP (1893) TransformationProof [EQUIVALENT, 0 ms] (1894) QDP (1895) DependencyGraphProof [EQUIVALENT, 0 ms] (1896) QDP (1897) TransformationProof [EQUIVALENT, 0 ms] (1898) QDP (1899) TransformationProof [EQUIVALENT, 0 ms] (1900) QDP (1901) TransformationProof [EQUIVALENT, 0 ms] (1902) QDP (1903) TransformationProof [EQUIVALENT, 0 ms] (1904) QDP (1905) TransformationProof [EQUIVALENT, 0 ms] (1906) QDP (1907) TransformationProof [EQUIVALENT, 0 ms] (1908) QDP (1909) TransformationProof [EQUIVALENT, 0 ms] (1910) QDP (1911) TransformationProof [EQUIVALENT, 0 ms] (1912) QDP (1913) TransformationProof [EQUIVALENT, 0 ms] (1914) QDP (1915) DependencyGraphProof [EQUIVALENT, 0 ms] (1916) QDP (1917) TransformationProof [EQUIVALENT, 0 ms] (1918) QDP (1919) TransformationProof [EQUIVALENT, 0 ms] (1920) QDP (1921) TransformationProof [EQUIVALENT, 0 ms] (1922) QDP (1923) DependencyGraphProof [EQUIVALENT, 0 ms] (1924) QDP (1925) TransformationProof [EQUIVALENT, 0 ms] (1926) QDP (1927) TransformationProof [EQUIVALENT, 0 ms] (1928) QDP (1929) TransformationProof [EQUIVALENT, 0 ms] (1930) QDP (1931) TransformationProof [EQUIVALENT, 0 ms] (1932) QDP (1933) TransformationProof [EQUIVALENT, 0 ms] (1934) QDP (1935) TransformationProof [EQUIVALENT, 0 ms] (1936) QDP (1937) TransformationProof [EQUIVALENT, 0 ms] (1938) QDP (1939) TransformationProof [EQUIVALENT, 0 ms] (1940) QDP (1941) TransformationProof [EQUIVALENT, 0 ms] (1942) QDP (1943) TransformationProof [EQUIVALENT, 0 ms] (1944) QDP (1945) TransformationProof [EQUIVALENT, 0 ms] (1946) QDP (1947) TransformationProof [EQUIVALENT, 0 ms] (1948) QDP (1949) DependencyGraphProof [EQUIVALENT, 0 ms] (1950) QDP (1951) TransformationProof [EQUIVALENT, 0 ms] (1952) QDP (1953) TransformationProof [EQUIVALENT, 0 ms] (1954) QDP (1955) TransformationProof [EQUIVALENT, 0 ms] (1956) QDP (1957) TransformationProof [EQUIVALENT, 0 ms] (1958) QDP (1959) TransformationProof [EQUIVALENT, 0 ms] (1960) QDP (1961) TransformationProof [EQUIVALENT, 0 ms] (1962) QDP (1963) TransformationProof [EQUIVALENT, 0 ms] (1964) QDP (1965) TransformationProof [EQUIVALENT, 0 ms] (1966) QDP (1967) TransformationProof [EQUIVALENT, 0 ms] (1968) QDP (1969) DependencyGraphProof [EQUIVALENT, 0 ms] (1970) QDP (1971) TransformationProof [EQUIVALENT, 0 ms] (1972) QDP (1973) TransformationProof [EQUIVALENT, 0 ms] (1974) QDP (1975) TransformationProof [EQUIVALENT, 0 ms] (1976) QDP (1977) DependencyGraphProof [EQUIVALENT, 0 ms] (1978) QDP (1979) TransformationProof [EQUIVALENT, 0 ms] (1980) QDP (1981) TransformationProof [EQUIVALENT, 0 ms] (1982) QDP (1983) TransformationProof [EQUIVALENT, 0 ms] (1984) QDP (1985) TransformationProof [EQUIVALENT, 0 ms] (1986) QDP (1987) TransformationProof [EQUIVALENT, 0 ms] (1988) QDP (1989) TransformationProof [EQUIVALENT, 0 ms] (1990) QDP (1991) TransformationProof [EQUIVALENT, 0 ms] (1992) QDP (1993) TransformationProof [EQUIVALENT, 0 ms] (1994) QDP (1995) TransformationProof [EQUIVALENT, 0 ms] (1996) QDP (1997) TransformationProof [EQUIVALENT, 0 ms] (1998) QDP (1999) TransformationProof [EQUIVALENT, 0 ms] (2000) QDP (2001) TransformationProof [EQUIVALENT, 0 ms] (2002) QDP (2003) TransformationProof [EQUIVALENT, 0 ms] (2004) QDP (2005) TransformationProof [EQUIVALENT, 0 ms] (2006) QDP (2007) TransformationProof [EQUIVALENT, 0 ms] (2008) QDP (2009) TransformationProof [EQUIVALENT, 0 ms] (2010) QDP (2011) TransformationProof [EQUIVALENT, 0 ms] (2012) QDP (2013) TransformationProof [EQUIVALENT, 0 ms] (2014) QDP (2015) TransformationProof [EQUIVALENT, 0 ms] (2016) QDP (2017) TransformationProof [EQUIVALENT, 0 ms] (2018) QDP (2019) TransformationProof [EQUIVALENT, 0 ms] (2020) QDP (2021) DependencyGraphProof [EQUIVALENT, 0 ms] (2022) QDP (2023) TransformationProof [EQUIVALENT, 0 ms] (2024) QDP (2025) DependencyGraphProof [EQUIVALENT, 0 ms] (2026) QDP (2027) TransformationProof [EQUIVALENT, 0 ms] (2028) QDP (2029) TransformationProof [EQUIVALENT, 0 ms] (2030) QDP (2031) TransformationProof [EQUIVALENT, 0 ms] (2032) QDP (2033) DependencyGraphProof [EQUIVALENT, 0 ms] (2034) QDP (2035) TransformationProof [EQUIVALENT, 0 ms] (2036) QDP (2037) TransformationProof [EQUIVALENT, 0 ms] (2038) QDP (2039) DependencyGraphProof [EQUIVALENT, 0 ms] (2040) QDP (2041) TransformationProof [EQUIVALENT, 0 ms] (2042) QDP (2043) DependencyGraphProof [EQUIVALENT, 0 ms] (2044) QDP (2045) TransformationProof [EQUIVALENT, 0 ms] (2046) QDP (2047) TransformationProof [EQUIVALENT, 0 ms] (2048) QDP (2049) DependencyGraphProof [EQUIVALENT, 0 ms] (2050) QDP (2051) TransformationProof [EQUIVALENT, 0 ms] (2052) QDP (2053) TransformationProof [EQUIVALENT, 0 ms] (2054) QDP (2055) TransformationProof [EQUIVALENT, 0 ms] (2056) QDP (2057) TransformationProof [EQUIVALENT, 0 ms] (2058) QDP (2059) TransformationProof [EQUIVALENT, 0 ms] (2060) QDP (2061) TransformationProof [EQUIVALENT, 0 ms] (2062) QDP (2063) DependencyGraphProof [EQUIVALENT, 0 ms] (2064) QDP (2065) TransformationProof [EQUIVALENT, 0 ms] (2066) QDP (2067) TransformationProof [EQUIVALENT, 0 ms] (2068) QDP (2069) TransformationProof [EQUIVALENT, 0 ms] (2070) QDP (2071) TransformationProof [EQUIVALENT, 0 ms] (2072) QDP (2073) DependencyGraphProof [EQUIVALENT, 0 ms] (2074) QDP (2075) TransformationProof [EQUIVALENT, 0 ms] (2076) QDP (2077) DependencyGraphProof [EQUIVALENT, 0 ms] (2078) QDP (2079) TransformationProof [EQUIVALENT, 0 ms] (2080) QDP (2081) TransformationProof [EQUIVALENT, 0 ms] (2082) QDP (2083) TransformationProof [EQUIVALENT, 0 ms] (2084) QDP (2085) DependencyGraphProof [EQUIVALENT, 0 ms] (2086) QDP (2087) TransformationProof [EQUIVALENT, 0 ms] (2088) QDP (2089) TransformationProof [EQUIVALENT, 0 ms] (2090) QDP (2091) TransformationProof [EQUIVALENT, 0 ms] (2092) QDP (2093) TransformationProof [EQUIVALENT, 0 ms] (2094) QDP (2095) TransformationProof [EQUIVALENT, 0 ms] (2096) QDP (2097) TransformationProof [EQUIVALENT, 0 ms] (2098) QDP (2099) DependencyGraphProof [EQUIVALENT, 0 ms] (2100) QDP (2101) TransformationProof [EQUIVALENT, 0 ms] (2102) QDP (2103) TransformationProof [EQUIVALENT, 0 ms] (2104) QDP (2105) TransformationProof [EQUIVALENT, 0 ms] (2106) QDP (2107) TransformationProof [EQUIVALENT, 0 ms] (2108) QDP (2109) DependencyGraphProof [EQUIVALENT, 0 ms] (2110) QDP (2111) TransformationProof [EQUIVALENT, 0 ms] (2112) QDP (2113) DependencyGraphProof [EQUIVALENT, 0 ms] (2114) QDP (2115) TransformationProof [EQUIVALENT, 0 ms] (2116) QDP (2117) TransformationProof [EQUIVALENT, 0 ms] (2118) QDP (2119) TransformationProof [EQUIVALENT, 0 ms] (2120) QDP (2121) TransformationProof [EQUIVALENT, 0 ms] (2122) QDP (2123) TransformationProof [EQUIVALENT, 0 ms] (2124) QDP (2125) TransformationProof [EQUIVALENT, 0 ms] (2126) QDP (2127) TransformationProof [EQUIVALENT, 0 ms] (2128) QDP (2129) TransformationProof [EQUIVALENT, 0 ms] (2130) QDP (2131) TransformationProof [EQUIVALENT, 0 ms] (2132) QDP (2133) TransformationProof [EQUIVALENT, 0 ms] (2134) QDP (2135) TransformationProof [EQUIVALENT, 0 ms] (2136) QDP (2137) TransformationProof [EQUIVALENT, 0 ms] (2138) QDP (2139) TransformationProof [EQUIVALENT, 0 ms] (2140) QDP (2141) TransformationProof [EQUIVALENT, 0 ms] (2142) QDP (2143) TransformationProof [EQUIVALENT, 0 ms] (2144) QDP (2145) TransformationProof [EQUIVALENT, 0 ms] (2146) QDP (2147) DependencyGraphProof [EQUIVALENT, 0 ms] (2148) QDP (2149) TransformationProof [EQUIVALENT, 0 ms] (2150) QDP (2151) TransformationProof [EQUIVALENT, 0 ms] (2152) QDP (2153) DependencyGraphProof [EQUIVALENT, 0 ms] (2154) QDP (2155) TransformationProof [EQUIVALENT, 0 ms] (2156) QDP (2157) DependencyGraphProof [EQUIVALENT, 0 ms] (2158) QDP (2159) TransformationProof [EQUIVALENT, 0 ms] (2160) QDP (2161) DependencyGraphProof [EQUIVALENT, 0 ms] (2162) QDP (2163) TransformationProof [EQUIVALENT, 0 ms] (2164) QDP (2165) DependencyGraphProof [EQUIVALENT, 0 ms] (2166) QDP (2167) TransformationProof [EQUIVALENT, 0 ms] (2168) QDP (2169) TransformationProof [EQUIVALENT, 0 ms] (2170) QDP (2171) TransformationProof [EQUIVALENT, 0 ms] (2172) QDP (2173) TransformationProof [EQUIVALENT, 0 ms] (2174) QDP (2175) DependencyGraphProof [EQUIVALENT, 0 ms] (2176) QDP (2177) TransformationProof [EQUIVALENT, 0 ms] (2178) QDP (2179) DependencyGraphProof [EQUIVALENT, 0 ms] (2180) QDP (2181) TransformationProof [EQUIVALENT, 0 ms] (2182) QDP (2183) DependencyGraphProof [EQUIVALENT, 0 ms] (2184) QDP (2185) TransformationProof [EQUIVALENT, 0 ms] (2186) QDP (2187) DependencyGraphProof [EQUIVALENT, 0 ms] (2188) QDP (2189) TransformationProof [EQUIVALENT, 0 ms] (2190) QDP (2191) TransformationProof [EQUIVALENT, 0 ms] (2192) QDP (2193) TransformationProof [EQUIVALENT, 0 ms] (2194) QDP (2195) TransformationProof [EQUIVALENT, 0 ms] (2196) QDP (2197) TransformationProof [EQUIVALENT, 0 ms] (2198) QDP (2199) TransformationProof [EQUIVALENT, 0 ms] (2200) QDP (2201) TransformationProof [EQUIVALENT, 0 ms] (2202) QDP (2203) DependencyGraphProof [EQUIVALENT, 0 ms] (2204) QDP (2205) TransformationProof [EQUIVALENT, 0 ms] (2206) QDP (2207) DependencyGraphProof [EQUIVALENT, 0 ms] (2208) QDP (2209) TransformationProof [EQUIVALENT, 0 ms] (2210) QDP (2211) TransformationProof [EQUIVALENT, 0 ms] (2212) QDP (2213) TransformationProof [EQUIVALENT, 0 ms] (2214) QDP (2215) TransformationProof [EQUIVALENT, 0 ms] (2216) QDP (2217) TransformationProof [EQUIVALENT, 0 ms] (2218) QDP (2219) TransformationProof [EQUIVALENT, 0 ms] (2220) QDP (2221) DependencyGraphProof [EQUIVALENT, 0 ms] (2222) QDP (2223) TransformationProof [EQUIVALENT, 0 ms] (2224) QDP (2225) DependencyGraphProof [EQUIVALENT, 0 ms] (2226) QDP (2227) TransformationProof [EQUIVALENT, 0 ms] (2228) QDP (2229) TransformationProof [EQUIVALENT, 0 ms] (2230) QDP (2231) TransformationProof [EQUIVALENT, 0 ms] (2232) QDP (2233) TransformationProof [EQUIVALENT, 0 ms] (2234) QDP (2235) DependencyGraphProof [EQUIVALENT, 0 ms] (2236) QDP (2237) TransformationProof [EQUIVALENT, 0 ms] (2238) QDP (2239) DependencyGraphProof [EQUIVALENT, 0 ms] (2240) QDP (2241) TransformationProof [EQUIVALENT, 0 ms] (2242) QDP (2243) TransformationProof [EQUIVALENT, 0 ms] (2244) QDP (2245) TransformationProof [EQUIVALENT, 0 ms] (2246) QDP (2247) TransformationProof [EQUIVALENT, 0 ms] (2248) QDP (2249) DependencyGraphProof [EQUIVALENT, 0 ms] (2250) QDP (2251) TransformationProof [EQUIVALENT, 0 ms] (2252) QDP (2253) DependencyGraphProof [EQUIVALENT, 0 ms] (2254) QDP (2255) TransformationProof [EQUIVALENT, 0 ms] (2256) QDP (2257) TransformationProof [EQUIVALENT, 0 ms] (2258) QDP (2259) DependencyGraphProof [EQUIVALENT, 0 ms] (2260) QDP (2261) TransformationProof [EQUIVALENT, 0 ms] (2262) QDP (2263) TransformationProof [EQUIVALENT, 0 ms] (2264) QDP (2265) TransformationProof [EQUIVALENT, 0 ms] (2266) QDP (2267) TransformationProof [EQUIVALENT, 0 ms] (2268) QDP (2269) TransformationProof [EQUIVALENT, 0 ms] (2270) QDP (2271) TransformationProof [EQUIVALENT, 0 ms] (2272) QDP (2273) DependencyGraphProof [EQUIVALENT, 0 ms] (2274) QDP (2275) TransformationProof [EQUIVALENT, 0 ms] (2276) QDP (2277) DependencyGraphProof [EQUIVALENT, 0 ms] (2278) QDP (2279) TransformationProof [EQUIVALENT, 1 ms] (2280) QDP (2281) TransformationProof [EQUIVALENT, 0 ms] (2282) QDP (2283) TransformationProof [EQUIVALENT, 0 ms] (2284) QDP (2285) TransformationProof [EQUIVALENT, 0 ms] (2286) QDP (2287) TransformationProof [EQUIVALENT, 0 ms] (2288) QDP (2289) DependencyGraphProof [EQUIVALENT, 0 ms] (2290) QDP (2291) TransformationProof [EQUIVALENT, 0 ms] (2292) QDP (2293) DependencyGraphProof [EQUIVALENT, 0 ms] (2294) QDP (2295) TransformationProof [EQUIVALENT, 0 ms] (2296) QDP (2297) DependencyGraphProof [EQUIVALENT, 0 ms] (2298) QDP (2299) TransformationProof [EQUIVALENT, 0 ms] (2300) QDP (2301) TransformationProof [EQUIVALENT, 0 ms] (2302) QDP (2303) DependencyGraphProof [EQUIVALENT, 0 ms] (2304) QDP (2305) TransformationProof [EQUIVALENT, 0 ms] (2306) QDP (2307) TransformationProof [EQUIVALENT, 0 ms] (2308) QDP (2309) TransformationProof [EQUIVALENT, 0 ms] (2310) QDP (2311) TransformationProof [EQUIVALENT, 0 ms] (2312) QDP (2313) TransformationProof [EQUIVALENT, 0 ms] (2314) QDP (2315) TransformationProof [EQUIVALENT, 0 ms] (2316) QDP (2317) TransformationProof [EQUIVALENT, 0 ms] (2318) QDP (2319) TransformationProof [EQUIVALENT, 0 ms] (2320) QDP (2321) DependencyGraphProof [EQUIVALENT, 0 ms] (2322) QDP (2323) TransformationProof [EQUIVALENT, 0 ms] (2324) QDP (2325) DependencyGraphProof [EQUIVALENT, 0 ms] (2326) QDP (2327) TransformationProof [EQUIVALENT, 0 ms] (2328) QDP (2329) TransformationProof [EQUIVALENT, 0 ms] (2330) QDP (2331) DependencyGraphProof [EQUIVALENT, 0 ms] (2332) QDP (2333) TransformationProof [EQUIVALENT, 0 ms] (2334) QDP (2335) TransformationProof [EQUIVALENT, 0 ms] (2336) QDP (2337) TransformationProof [EQUIVALENT, 0 ms] (2338) QDP (2339) TransformationProof [EQUIVALENT, 0 ms] (2340) QDP (2341) TransformationProof [EQUIVALENT, 0 ms] (2342) QDP (2343) TransformationProof [EQUIVALENT, 0 ms] (2344) QDP (2345) DependencyGraphProof [EQUIVALENT, 0 ms] (2346) QDP (2347) TransformationProof [EQUIVALENT, 0 ms] (2348) QDP (2349) DependencyGraphProof [EQUIVALENT, 0 ms] (2350) QDP (2351) TransformationProof [EQUIVALENT, 0 ms] (2352) QDP (2353) DependencyGraphProof [EQUIVALENT, 0 ms] (2354) QDP (2355) TransformationProof [EQUIVALENT, 0 ms] (2356) QDP (2357) TransformationProof [EQUIVALENT, 0 ms] (2358) QDP (2359) TransformationProof [EQUIVALENT, 0 ms] (2360) QDP (2361) TransformationProof [EQUIVALENT, 0 ms] (2362) QDP (2363) TransformationProof [EQUIVALENT, 0 ms] (2364) QDP (2365) TransformationProof [EQUIVALENT, 0 ms] (2366) QDP (2367) TransformationProof [EQUIVALENT, 0 ms] (2368) QDP (2369) DependencyGraphProof [EQUIVALENT, 0 ms] (2370) QDP (2371) TransformationProof [EQUIVALENT, 0 ms] (2372) QDP (2373) DependencyGraphProof [EQUIVALENT, 0 ms] (2374) QDP (2375) TransformationProof [EQUIVALENT, 0 ms] (2376) QDP (2377) TransformationProof [EQUIVALENT, 0 ms] (2378) QDP (2379) TransformationProof [EQUIVALENT, 0 ms] (2380) QDP (2381) DependencyGraphProof [EQUIVALENT, 0 ms] (2382) QDP (2383) TransformationProof [EQUIVALENT, 0 ms] (2384) QDP (2385) TransformationProof [EQUIVALENT, 0 ms] (2386) QDP (2387) DependencyGraphProof [EQUIVALENT, 0 ms] (2388) QDP (2389) TransformationProof [EQUIVALENT, 0 ms] (2390) QDP (2391) DependencyGraphProof [EQUIVALENT, 0 ms] (2392) QDP (2393) TransformationProof [EQUIVALENT, 0 ms] (2394) QDP (2395) TransformationProof [EQUIVALENT, 0 ms] (2396) QDP (2397) DependencyGraphProof [EQUIVALENT, 0 ms] (2398) QDP (2399) TransformationProof [EQUIVALENT, 0 ms] (2400) QDP (2401) TransformationProof [EQUIVALENT, 0 ms] (2402) QDP (2403) TransformationProof [EQUIVALENT, 0 ms] (2404) QDP (2405) TransformationProof [EQUIVALENT, 0 ms] (2406) QDP (2407) TransformationProof [EQUIVALENT, 0 ms] (2408) QDP (2409) DependencyGraphProof [EQUIVALENT, 0 ms] (2410) QDP (2411) TransformationProof [EQUIVALENT, 0 ms] (2412) QDP (2413) TransformationProof [EQUIVALENT, 0 ms] (2414) QDP (2415) TransformationProof [EQUIVALENT, 0 ms] (2416) QDP (2417) TransformationProof [EQUIVALENT, 0 ms] (2418) QDP (2419) TransformationProof [EQUIVALENT, 0 ms] (2420) QDP (2421) TransformationProof [EQUIVALENT, 0 ms] (2422) QDP (2423) DependencyGraphProof [EQUIVALENT, 0 ms] (2424) QDP (2425) TransformationProof [EQUIVALENT, 0 ms] (2426) QDP (2427) TransformationProof [EQUIVALENT, 0 ms] (2428) QDP (2429) TransformationProof [EQUIVALENT, 0 ms] (2430) QDP (2431) TransformationProof [EQUIVALENT, 0 ms] (2432) QDP (2433) DependencyGraphProof [EQUIVALENT, 0 ms] (2434) QDP (2435) TransformationProof [EQUIVALENT, 0 ms] (2436) QDP (2437) DependencyGraphProof [EQUIVALENT, 0 ms] (2438) QDP (2439) TransformationProof [EQUIVALENT, 0 ms] (2440) QDP (2441) TransformationProof [EQUIVALENT, 0 ms] (2442) QDP (2443) TransformationProof [EQUIVALENT, 0 ms] (2444) QDP (2445) TransformationProof [EQUIVALENT, 0 ms] (2446) QDP (2447) TransformationProof [EQUIVALENT, 0 ms] (2448) QDP (2449) TransformationProof [EQUIVALENT, 0 ms] (2450) QDP (2451) TransformationProof [EQUIVALENT, 0 ms] (2452) QDP (2453) TransformationProof [EQUIVALENT, 0 ms] (2454) QDP (2455) TransformationProof [EQUIVALENT, 0 ms] (2456) QDP (2457) TransformationProof [EQUIVALENT, 0 ms] (2458) QDP (2459) TransformationProof [EQUIVALENT, 0 ms] (2460) QDP (2461) TransformationProof [EQUIVALENT, 0 ms] (2462) QDP (2463) TransformationProof [EQUIVALENT, 0 ms] (2464) QDP (2465) TransformationProof [EQUIVALENT, 0 ms] (2466) QDP (2467) TransformationProof [EQUIVALENT, 0 ms] (2468) QDP (2469) TransformationProof [EQUIVALENT, 0 ms] (2470) QDP (2471) TransformationProof [EQUIVALENT, 0 ms] (2472) QDP (2473) TransformationProof [EQUIVALENT, 0 ms] (2474) QDP (2475) TransformationProof [EQUIVALENT, 0 ms] (2476) QDP (2477) DependencyGraphProof [EQUIVALENT, 0 ms] (2478) QDP (2479) TransformationProof [EQUIVALENT, 1 ms] (2480) QDP (2481) TransformationProof [EQUIVALENT, 0 ms] (2482) QDP (2483) DependencyGraphProof [EQUIVALENT, 0 ms] (2484) QDP (2485) TransformationProof [EQUIVALENT, 0 ms] (2486) QDP (2487) TransformationProof [EQUIVALENT, 0 ms] (2488) QDP (2489) TransformationProof [EQUIVALENT, 0 ms] (2490) QDP (2491) TransformationProof [EQUIVALENT, 0 ms] (2492) QDP (2493) TransformationProof [EQUIVALENT, 0 ms] (2494) QDP (2495) TransformationProof [EQUIVALENT, 0 ms] (2496) QDP (2497) DependencyGraphProof [EQUIVALENT, 0 ms] (2498) QDP (2499) TransformationProof [EQUIVALENT, 0 ms] (2500) QDP (2501) DependencyGraphProof [EQUIVALENT, 0 ms] (2502) QDP (2503) TransformationProof [EQUIVALENT, 0 ms] (2504) QDP (2505) TransformationProof [EQUIVALENT, 0 ms] (2506) QDP (2507) DependencyGraphProof [EQUIVALENT, 0 ms] (2508) QDP (2509) TransformationProof [EQUIVALENT, 0 ms] (2510) QDP (2511) TransformationProof [EQUIVALENT, 0 ms] (2512) QDP (2513) TransformationProof [EQUIVALENT, 0 ms] (2514) QDP (2515) TransformationProof [EQUIVALENT, 0 ms] (2516) QDP (2517) TransformationProof [EQUIVALENT, 0 ms] (2518) QDP (2519) TransformationProof [EQUIVALENT, 0 ms] (2520) QDP (2521) DependencyGraphProof [EQUIVALENT, 0 ms] (2522) QDP (2523) TransformationProof [EQUIVALENT, 0 ms] (2524) QDP (2525) DependencyGraphProof [EQUIVALENT, 0 ms] (2526) QDP (2527) TransformationProof [EQUIVALENT, 0 ms] (2528) QDP (2529) TransformationProof [EQUIVALENT, 0 ms] (2530) QDP (2531) TransformationProof [EQUIVALENT, 0 ms] (2532) QDP (2533) TransformationProof [EQUIVALENT, 0 ms] (2534) QDP (2535) DependencyGraphProof [EQUIVALENT, 0 ms] (2536) QDP (2537) TransformationProof [EQUIVALENT, 0 ms] (2538) QDP (2539) DependencyGraphProof [EQUIVALENT, 0 ms] (2540) QDP (2541) TransformationProof [EQUIVALENT, 0 ms] (2542) QDP (2543) DependencyGraphProof [EQUIVALENT, 0 ms] (2544) QDP (2545) TransformationProof [EQUIVALENT, 0 ms] (2546) QDP (2547) DependencyGraphProof [EQUIVALENT, 0 ms] (2548) QDP (2549) TransformationProof [EQUIVALENT, 0 ms] (2550) QDP (2551) TransformationProof [EQUIVALENT, 0 ms] (2552) QDP (2553) TransformationProof [EQUIVALENT, 0 ms] (2554) QDP (2555) TransformationProof [EQUIVALENT, 0 ms] (2556) QDP (2557) TransformationProof [EQUIVALENT, 0 ms] (2558) QDP (2559) DependencyGraphProof [EQUIVALENT, 0 ms] (2560) QDP (2561) TransformationProof [EQUIVALENT, 0 ms] (2562) QDP (2563) DependencyGraphProof [EQUIVALENT, 0 ms] (2564) QDP (2565) TransformationProof [EQUIVALENT, 0 ms] (2566) QDP (2567) DependencyGraphProof [EQUIVALENT, 0 ms] (2568) QDP (2569) TransformationProof [EQUIVALENT, 0 ms] (2570) QDP (2571) TransformationProof [EQUIVALENT, 0 ms] (2572) QDP (2573) TransformationProof [EQUIVALENT, 0 ms] (2574) QDP (2575) TransformationProof [EQUIVALENT, 0 ms] (2576) QDP (2577) TransformationProof [EQUIVALENT, 0 ms] (2578) QDP (2579) DependencyGraphProof [EQUIVALENT, 0 ms] (2580) QDP (2581) TransformationProof [EQUIVALENT, 0 ms] (2582) QDP (2583) DependencyGraphProof [EQUIVALENT, 0 ms] (2584) QDP (2585) TransformationProof [EQUIVALENT, 0 ms] (2586) QDP (2587) DependencyGraphProof [EQUIVALENT, 0 ms] (2588) QDP (2589) TransformationProof [EQUIVALENT, 0 ms] (2590) QDP (2591) TransformationProof [EQUIVALENT, 0 ms] (2592) QDP (2593) TransformationProof [EQUIVALENT, 0 ms] (2594) QDP (2595) TransformationProof [EQUIVALENT, 0 ms] (2596) QDP (2597) TransformationProof [EQUIVALENT, 0 ms] (2598) QDP (2599) TransformationProof [EQUIVALENT, 0 ms] (2600) QDP (2601) DependencyGraphProof [EQUIVALENT, 0 ms] (2602) QDP (2603) TransformationProof [EQUIVALENT, 0 ms] (2604) QDP (2605) DependencyGraphProof [EQUIVALENT, 0 ms] (2606) QDP (2607) TransformationProof [EQUIVALENT, 0 ms] (2608) QDP (2609) DependencyGraphProof [EQUIVALENT, 0 ms] (2610) QDP (2611) TransformationProof [EQUIVALENT, 0 ms] (2612) QDP (2613) TransformationProof [EQUIVALENT, 0 ms] (2614) QDP (2615) DependencyGraphProof [EQUIVALENT, 0 ms] (2616) QDP (2617) TransformationProof [EQUIVALENT, 0 ms] (2618) QDP (2619) TransformationProof [EQUIVALENT, 0 ms] (2620) QDP (2621) DependencyGraphProof [EQUIVALENT, 0 ms] (2622) QDP (2623) TransformationProof [EQUIVALENT, 0 ms] (2624) QDP (2625) TransformationProof [EQUIVALENT, 0 ms] (2626) QDP (2627) TransformationProof [EQUIVALENT, 0 ms] (2628) QDP (2629) DependencyGraphProof [EQUIVALENT, 0 ms] (2630) QDP (2631) TransformationProof [EQUIVALENT, 0 ms] (2632) QDP (2633) DependencyGraphProof [EQUIVALENT, 0 ms] (2634) QDP (2635) TransformationProof [EQUIVALENT, 0 ms] (2636) QDP (2637) DependencyGraphProof [EQUIVALENT, 0 ms] (2638) QDP (2639) TransformationProof [EQUIVALENT, 0 ms] (2640) QDP (2641) TransformationProof [EQUIVALENT, 0 ms] (2642) QDP (2643) DependencyGraphProof [EQUIVALENT, 0 ms] (2644) QDP (2645) TransformationProof [EQUIVALENT, 0 ms] (2646) QDP (2647) DependencyGraphProof [EQUIVALENT, 0 ms] (2648) QDP (2649) TransformationProof [EQUIVALENT, 0 ms] (2650) QDP (2651) TransformationProof [EQUIVALENT, 0 ms] (2652) QDP (2653) TransformationProof [EQUIVALENT, 0 ms] (2654) QDP (2655) DependencyGraphProof [EQUIVALENT, 0 ms] (2656) QDP (2657) TransformationProof [EQUIVALENT, 0 ms] (2658) QDP (2659) DependencyGraphProof [EQUIVALENT, 0 ms] (2660) QDP (2661) TransformationProof [EQUIVALENT, 0 ms] (2662) QDP (2663) DependencyGraphProof [EQUIVALENT, 0 ms] (2664) QDP (2665) TransformationProof [EQUIVALENT, 0 ms] (2666) QDP (2667) DependencyGraphProof [EQUIVALENT, 0 ms] (2668) QDP (2669) TransformationProof [EQUIVALENT, 0 ms] (2670) QDP (2671) TransformationProof [EQUIVALENT, 0 ms] (2672) QDP (2673) DependencyGraphProof [EQUIVALENT, 0 ms] (2674) QDP (2675) TransformationProof [EQUIVALENT, 0 ms] (2676) QDP (2677) TransformationProof [EQUIVALENT, 0 ms] (2678) QDP (2679) TransformationProof [EQUIVALENT, 0 ms] (2680) QDP (2681) DependencyGraphProof [EQUIVALENT, 0 ms] (2682) QDP (2683) TransformationProof [EQUIVALENT, 0 ms] (2684) QDP (2685) DependencyGraphProof [EQUIVALENT, 0 ms] (2686) QDP (2687) TransformationProof [EQUIVALENT, 0 ms] (2688) QDP (2689) TransformationProof [EQUIVALENT, 0 ms] (2690) QDP (2691) TransformationProof [EQUIVALENT, 0 ms] (2692) QDP (2693) DependencyGraphProof [EQUIVALENT, 0 ms] (2694) QDP (2695) TransformationProof [EQUIVALENT, 0 ms] (2696) QDP (2697) DependencyGraphProof [EQUIVALENT, 0 ms] (2698) QDP (2699) TransformationProof [EQUIVALENT, 0 ms] (2700) QDP (2701) TransformationProof [EQUIVALENT, 0 ms] (2702) QDP (2703) TransformationProof [EQUIVALENT, 0 ms] (2704) QDP (2705) TransformationProof [EQUIVALENT, 0 ms] (2706) QDP (2707) TransformationProof [EQUIVALENT, 0 ms] (2708) QDP (2709) TransformationProof [EQUIVALENT, 0 ms] (2710) QDP (2711) DependencyGraphProof [EQUIVALENT, 0 ms] (2712) QDP (2713) TransformationProof [EQUIVALENT, 0 ms] (2714) QDP (2715) DependencyGraphProof [EQUIVALENT, 0 ms] (2716) QDP (2717) TransformationProof [EQUIVALENT, 0 ms] (2718) QDP (2719) DependencyGraphProof [EQUIVALENT, 0 ms] (2720) QDP (2721) TransformationProof [EQUIVALENT, 0 ms] (2722) QDP (2723) TransformationProof [EQUIVALENT, 0 ms] (2724) QDP (2725) TransformationProof [EQUIVALENT, 0 ms] (2726) QDP (2727) TransformationProof [EQUIVALENT, 0 ms] (2728) QDP (2729) TransformationProof [EQUIVALENT, 0 ms] (2730) QDP (2731) TransformationProof [EQUIVALENT, 0 ms] (2732) QDP (2733) DependencyGraphProof [EQUIVALENT, 0 ms] (2734) QDP (2735) TransformationProof [EQUIVALENT, 0 ms] (2736) QDP (2737) TransformationProof [EQUIVALENT, 0 ms] (2738) QDP (2739) TransformationProof [EQUIVALENT, 0 ms] (2740) QDP (2741) DependencyGraphProof [EQUIVALENT, 0 ms] (2742) QDP (2743) TransformationProof [EQUIVALENT, 0 ms] (2744) QDP (2745) DependencyGraphProof [EQUIVALENT, 0 ms] (2746) QDP (2747) TransformationProof [EQUIVALENT, 0 ms] (2748) QDP (2749) TransformationProof [EQUIVALENT, 0 ms] (2750) QDP (2751) TransformationProof [EQUIVALENT, 0 ms] (2752) QDP (2753) DependencyGraphProof [EQUIVALENT, 0 ms] (2754) QDP (2755) TransformationProof [EQUIVALENT, 0 ms] (2756) QDP (2757) TransformationProof [EQUIVALENT, 0 ms] (2758) QDP (2759) TransformationProof [EQUIVALENT, 0 ms] (2760) QDP (2761) DependencyGraphProof [EQUIVALENT, 0 ms] (2762) QDP (2763) TransformationProof [EQUIVALENT, 0 ms] (2764) QDP (2765) DependencyGraphProof [EQUIVALENT, 0 ms] (2766) QDP (2767) TransformationProof [EQUIVALENT, 0 ms] (2768) QDP (2769) DependencyGraphProof [EQUIVALENT, 0 ms] (2770) QDP (2771) TransformationProof [EQUIVALENT, 0 ms] (2772) QDP (2773) DependencyGraphProof [EQUIVALENT, 0 ms] (2774) QDP (2775) TransformationProof [EQUIVALENT, 0 ms] (2776) QDP (2777) TransformationProof [EQUIVALENT, 0 ms] (2778) QDP (2779) DependencyGraphProof [EQUIVALENT, 0 ms] (2780) QDP (2781) TransformationProof [EQUIVALENT, 0 ms] (2782) QDP (2783) DependencyGraphProof [EQUIVALENT, 0 ms] (2784) QDP (2785) TransformationProof [EQUIVALENT, 0 ms] (2786) QDP (2787) DependencyGraphProof [EQUIVALENT, 0 ms] (2788) QDP (2789) TransformationProof [EQUIVALENT, 0 ms] (2790) QDP (2791) TransformationProof [EQUIVALENT, 0 ms] (2792) QDP (2793) DependencyGraphProof [EQUIVALENT, 0 ms] (2794) QDP (2795) TransformationProof [EQUIVALENT, 0 ms] (2796) QDP (2797) TransformationProof [EQUIVALENT, 0 ms] (2798) QDP (2799) UsableRulesReductionPairsProof [EQUIVALENT, 7 ms] (2800) QDP (2801) QDPOrderProof [EQUIVALENT, 14 ms] (2802) QDP (2803) QDPOrderProof [EQUIVALENT, 10 ms] (2804) QDP (2805) QDPOrderProof [EQUIVALENT, 12 ms] (2806) QDP (2807) NonTerminationLoopProof [COMPLETE, 882 ms] (2808) NO ---------------------------------------- (0) Obligation: Conditional term rewrite system: The TRS R consists of the following rules: a -> c b -> c a -> d b -> d c -> e c -> k d -> k A -> h(f(a), f(b)) h(x, x) -> g(x, x, f(k)) g(d, x, x) -> A The conditional TRS C consists of the following conditional rules: f(x) -> x <= x -> e ---------------------------------------- (1) CTRSToQTRSProof (SOUND) The conditional rules have been transormed into unconditional rules according to [CTRS,AAECCNOC]. ---------------------------------------- (2) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c b -> c a -> d b -> d c -> e c -> k d -> k A -> h(f(a), f(b)) h(x, x) -> g(x, x, f(k)) g(d, x, x) -> A Q is empty. ---------------------------------------- (3) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (4) Obligation: Q DP problem: The TRS P consists of the following rules: F(x) -> U1^1(x, x) A^1 -> C B -> C A^1 -> D B -> D A^2 -> H(f(a), f(b)) A^2 -> F(a) A^2 -> A^1 A^2 -> F(b) A^2 -> B H(x, x) -> G(x, x, f(k)) H(x, x) -> F(k) G(d, x, x) -> A^2 The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c b -> c a -> d b -> d c -> e c -> k d -> k A -> h(f(a), f(b)) h(x, x) -> g(x, x, f(k)) g(d, x, x) -> A Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (5) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 10 less nodes. ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(a), f(b)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c b -> c a -> d b -> d c -> e c -> k d -> k A -> h(f(a), f(b)) h(x, x) -> g(x, x, f(k)) g(d, x, x) -> A Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: A -> h(f(a), f(b)) h(x, x) -> g(x, x, f(k)) g(d, x, x) -> A Used ordering: POLO with Polynomial interpretation [POLO]: POL(A^2) = 0 POL(G(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(H(x_1, x_2)) = 2*x_1 + 2*x_2 POL(U1(x_1, x_2)) = x_1 + x_2 POL(a) = 0 POL(b) = 0 POL(c) = 0 POL(d) = 0 POL(e) = 0 POL(f(x_1)) = 2*x_1 POL(k) = 0 ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(a), f(b)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), f(b)),A^2 -> H(U1(a, a), f(b))) (A^2 -> H(f(c), f(b)),A^2 -> H(f(c), f(b))) (A^2 -> H(f(d), f(b)),A^2 -> H(f(d), f(b))) (A^2 -> H(f(a), U1(b, b)),A^2 -> H(f(a), U1(b, b))) (A^2 -> H(f(a), f(c)),A^2 -> H(f(a), f(c))) (A^2 -> H(f(a), f(d)),A^2 -> H(f(a), f(d))) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(a, a), f(b)) A^2 -> H(f(c), f(b)) A^2 -> H(f(d), f(b)) A^2 -> H(f(a), U1(b, b)) A^2 -> H(f(a), f(c)) A^2 -> H(f(a), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), f(b)),A^2 -> H(U1(c, a), f(b))) (A^2 -> H(U1(d, a), f(b)),A^2 -> H(U1(d, a), f(b))) (A^2 -> H(U1(a, c), f(b)),A^2 -> H(U1(a, c), f(b))) (A^2 -> H(U1(a, d), f(b)),A^2 -> H(U1(a, d), f(b))) (A^2 -> H(U1(a, a), U1(b, b)),A^2 -> H(U1(a, a), U1(b, b))) (A^2 -> H(U1(a, a), f(c)),A^2 -> H(U1(a, a), f(c))) (A^2 -> H(U1(a, a), f(d)),A^2 -> H(U1(a, a), f(d))) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(b)) A^2 -> H(f(d), f(b)) A^2 -> H(f(a), U1(b, b)) A^2 -> H(f(a), f(c)) A^2 -> H(f(a), f(d)) A^2 -> H(U1(c, a), f(b)) A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(b)),A^2 -> H(U1(c, c), f(b))) (A^2 -> H(f(e), f(b)),A^2 -> H(f(e), f(b))) (A^2 -> H(f(k), f(b)),A^2 -> H(f(k), f(b))) (A^2 -> H(f(c), U1(b, b)),A^2 -> H(f(c), U1(b, b))) (A^2 -> H(f(c), f(c)),A^2 -> H(f(c), f(c))) (A^2 -> H(f(c), f(d)),A^2 -> H(f(c), f(d))) ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(b)) A^2 -> H(f(a), U1(b, b)) A^2 -> H(f(a), f(c)) A^2 -> H(f(a), f(d)) A^2 -> H(U1(c, a), f(b)) A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), f(b)),A^2 -> H(U1(d, d), f(b))) (A^2 -> H(f(k), f(b)),A^2 -> H(f(k), f(b))) (A^2 -> H(f(d), U1(b, b)),A^2 -> H(f(d), U1(b, b))) (A^2 -> H(f(d), f(c)),A^2 -> H(f(d), f(c))) (A^2 -> H(f(d), f(d)),A^2 -> H(f(d), f(d))) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(a), U1(b, b)) A^2 -> H(f(a), f(c)) A^2 -> H(f(a), f(d)) A^2 -> H(U1(c, a), f(b)) A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(b, b)),A^2 -> H(U1(a, a), U1(b, b))) (A^2 -> H(f(c), U1(b, b)),A^2 -> H(f(c), U1(b, b))) (A^2 -> H(f(d), U1(b, b)),A^2 -> H(f(d), U1(b, b))) (A^2 -> H(f(a), U1(c, b)),A^2 -> H(f(a), U1(c, b))) (A^2 -> H(f(a), U1(d, b)),A^2 -> H(f(a), U1(d, b))) (A^2 -> H(f(a), U1(b, c)),A^2 -> H(f(a), U1(b, c))) (A^2 -> H(f(a), U1(b, d)),A^2 -> H(f(a), U1(b, d))) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(a), f(c)) A^2 -> H(f(a), f(d)) A^2 -> H(U1(c, a), f(b)) A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), f(c)),A^2 -> H(U1(a, a), f(c))) (A^2 -> H(f(c), f(c)),A^2 -> H(f(c), f(c))) (A^2 -> H(f(d), f(c)),A^2 -> H(f(d), f(c))) (A^2 -> H(f(a), U1(c, c)),A^2 -> H(f(a), U1(c, c))) (A^2 -> H(f(a), f(e)),A^2 -> H(f(a), f(e))) (A^2 -> H(f(a), f(k)),A^2 -> H(f(a), f(k))) ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(a), f(d)) A^2 -> H(U1(c, a), f(b)) A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), f(d)),A^2 -> H(U1(a, a), f(d))) (A^2 -> H(f(c), f(d)),A^2 -> H(f(c), f(d))) (A^2 -> H(f(d), f(d)),A^2 -> H(f(d), f(d))) (A^2 -> H(f(a), U1(d, d)),A^2 -> H(f(a), U1(d, d))) (A^2 -> H(f(a), f(k)),A^2 -> H(f(a), f(k))) ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(c, a), f(b)) A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), f(b)),A^2 -> H(U1(e, a), f(b))) (A^2 -> H(U1(k, a), f(b)),A^2 -> H(U1(k, a), f(b))) (A^2 -> H(U1(c, c), f(b)),A^2 -> H(U1(c, c), f(b))) (A^2 -> H(U1(c, d), f(b)),A^2 -> H(U1(c, d), f(b))) (A^2 -> H(U1(c, a), U1(b, b)),A^2 -> H(U1(c, a), U1(b, b))) (A^2 -> H(U1(c, a), f(c)),A^2 -> H(U1(c, a), f(c))) (A^2 -> H(U1(c, a), f(d)),A^2 -> H(U1(c, a), f(d))) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(d, a), f(b)) A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), f(b)),A^2 -> H(U1(k, a), f(b))) (A^2 -> H(U1(d, c), f(b)),A^2 -> H(U1(d, c), f(b))) (A^2 -> H(U1(d, d), f(b)),A^2 -> H(U1(d, d), f(b))) (A^2 -> H(U1(d, a), U1(b, b)),A^2 -> H(U1(d, a), U1(b, b))) (A^2 -> H(U1(d, a), f(c)),A^2 -> H(U1(d, a), f(c))) (A^2 -> H(U1(d, a), f(d)),A^2 -> H(U1(d, a), f(d))) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(a, c), f(b)) A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(b)),A^2 -> H(U1(c, c), f(b))) (A^2 -> H(U1(d, c), f(b)),A^2 -> H(U1(d, c), f(b))) (A^2 -> H(U1(a, e), f(b)),A^2 -> H(U1(a, e), f(b))) (A^2 -> H(U1(a, k), f(b)),A^2 -> H(U1(a, k), f(b))) (A^2 -> H(U1(a, c), U1(b, b)),A^2 -> H(U1(a, c), U1(b, b))) (A^2 -> H(U1(a, c), f(c)),A^2 -> H(U1(a, c), f(c))) (A^2 -> H(U1(a, c), f(d)),A^2 -> H(U1(a, c), f(d))) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(a, d), f(b)) A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), f(b)),A^2 -> H(U1(c, d), f(b))) (A^2 -> H(U1(d, d), f(b)),A^2 -> H(U1(d, d), f(b))) (A^2 -> H(U1(a, k), f(b)),A^2 -> H(U1(a, k), f(b))) (A^2 -> H(U1(a, d), U1(b, b)),A^2 -> H(U1(a, d), U1(b, b))) (A^2 -> H(U1(a, d), f(c)),A^2 -> H(U1(a, d), f(c))) (A^2 -> H(U1(a, d), f(d)),A^2 -> H(U1(a, d), f(d))) ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(a, a), U1(b, b)) A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(b, b)),A^2 -> H(U1(c, a), U1(b, b))) (A^2 -> H(U1(d, a), U1(b, b)),A^2 -> H(U1(d, a), U1(b, b))) (A^2 -> H(U1(a, c), U1(b, b)),A^2 -> H(U1(a, c), U1(b, b))) (A^2 -> H(U1(a, d), U1(b, b)),A^2 -> H(U1(a, d), U1(b, b))) (A^2 -> H(U1(a, a), U1(c, b)),A^2 -> H(U1(a, a), U1(c, b))) (A^2 -> H(U1(a, a), U1(d, b)),A^2 -> H(U1(a, a), U1(d, b))) (A^2 -> H(U1(a, a), U1(b, c)),A^2 -> H(U1(a, a), U1(b, c))) (A^2 -> H(U1(a, a), U1(b, d)),A^2 -> H(U1(a, a), U1(b, d))) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(a, a), f(c)) A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), f(c)),A^2 -> H(U1(c, a), f(c))) (A^2 -> H(U1(d, a), f(c)),A^2 -> H(U1(d, a), f(c))) (A^2 -> H(U1(a, c), f(c)),A^2 -> H(U1(a, c), f(c))) (A^2 -> H(U1(a, d), f(c)),A^2 -> H(U1(a, d), f(c))) (A^2 -> H(U1(a, a), U1(c, c)),A^2 -> H(U1(a, a), U1(c, c))) (A^2 -> H(U1(a, a), f(e)),A^2 -> H(U1(a, a), f(e))) (A^2 -> H(U1(a, a), f(k)),A^2 -> H(U1(a, a), f(k))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(a, a), f(d)) A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), f(d)),A^2 -> H(U1(c, a), f(d))) (A^2 -> H(U1(d, a), f(d)),A^2 -> H(U1(d, a), f(d))) (A^2 -> H(U1(a, c), f(d)),A^2 -> H(U1(a, c), f(d))) (A^2 -> H(U1(a, d), f(d)),A^2 -> H(U1(a, d), f(d))) (A^2 -> H(U1(a, a), U1(d, d)),A^2 -> H(U1(a, a), U1(d, d))) (A^2 -> H(U1(a, a), f(k)),A^2 -> H(U1(a, a), f(k))) ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(U1(c, c), f(b)) A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), f(b)),A^2 -> H(U1(e, c), f(b))) (A^2 -> H(U1(k, c), f(b)),A^2 -> H(U1(k, c), f(b))) (A^2 -> H(U1(c, e), f(b)),A^2 -> H(U1(c, e), f(b))) (A^2 -> H(U1(c, k), f(b)),A^2 -> H(U1(c, k), f(b))) (A^2 -> H(U1(c, c), U1(b, b)),A^2 -> H(U1(c, c), U1(b, b))) (A^2 -> H(U1(c, c), f(c)),A^2 -> H(U1(c, c), f(c))) (A^2 -> H(U1(c, c), f(d)),A^2 -> H(U1(c, c), f(d))) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(e), f(b)) A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(b)),A^2 -> H(U1(e, e), f(b))) (A^2 -> H(f(e), U1(b, b)),A^2 -> H(f(e), U1(b, b))) (A^2 -> H(f(e), f(c)),A^2 -> H(f(e), f(c))) (A^2 -> H(f(e), f(d)),A^2 -> H(f(e), f(d))) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(k), f(b)) A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(b)),A^2 -> H(U1(k, k), f(b))) (A^2 -> H(f(k), U1(b, b)),A^2 -> H(f(k), U1(b, b))) (A^2 -> H(f(k), f(c)),A^2 -> H(f(k), f(c))) (A^2 -> H(f(k), f(d)),A^2 -> H(f(k), f(d))) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), U1(b, b)) A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, b)),A^2 -> H(U1(c, c), U1(b, b))) (A^2 -> H(f(e), U1(b, b)),A^2 -> H(f(e), U1(b, b))) (A^2 -> H(f(k), U1(b, b)),A^2 -> H(f(k), U1(b, b))) (A^2 -> H(f(c), U1(c, b)),A^2 -> H(f(c), U1(c, b))) (A^2 -> H(f(c), U1(d, b)),A^2 -> H(f(c), U1(d, b))) (A^2 -> H(f(c), U1(b, c)),A^2 -> H(f(c), U1(b, c))) (A^2 -> H(f(c), U1(b, d)),A^2 -> H(f(c), U1(b, d))) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(c), f(d)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(d)),A^2 -> H(U1(c, c), f(d))) (A^2 -> H(f(e), f(d)),A^2 -> H(f(e), f(d))) (A^2 -> H(f(k), f(d)),A^2 -> H(f(k), f(d))) (A^2 -> H(f(c), U1(d, d)),A^2 -> H(f(c), U1(d, d))) (A^2 -> H(f(c), f(k)),A^2 -> H(f(c), f(k))) ---------------------------------------- (46) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(U1(d, d), f(b)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (47) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), f(b)),A^2 -> H(U1(k, d), f(b))) (A^2 -> H(U1(d, k), f(b)),A^2 -> H(U1(d, k), f(b))) (A^2 -> H(U1(d, d), U1(b, b)),A^2 -> H(U1(d, d), U1(b, b))) (A^2 -> H(U1(d, d), f(c)),A^2 -> H(U1(d, d), f(c))) (A^2 -> H(U1(d, d), f(d)),A^2 -> H(U1(d, d), f(d))) ---------------------------------------- (48) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), U1(b, b)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (49) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(b, b)),A^2 -> H(U1(d, d), U1(b, b))) (A^2 -> H(f(k), U1(b, b)),A^2 -> H(f(k), U1(b, b))) (A^2 -> H(f(d), U1(c, b)),A^2 -> H(f(d), U1(c, b))) (A^2 -> H(f(d), U1(d, b)),A^2 -> H(f(d), U1(d, b))) (A^2 -> H(f(d), U1(b, c)),A^2 -> H(f(d), U1(b, c))) (A^2 -> H(f(d), U1(b, d)),A^2 -> H(f(d), U1(b, d))) ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), f(c)),A^2 -> H(U1(d, d), f(c))) (A^2 -> H(f(k), f(c)),A^2 -> H(f(k), f(c))) (A^2 -> H(f(d), U1(c, c)),A^2 -> H(f(d), U1(c, c))) (A^2 -> H(f(d), f(e)),A^2 -> H(f(d), f(e))) (A^2 -> H(f(d), f(k)),A^2 -> H(f(d), f(k))) ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, b)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(c, b)),A^2 -> H(U1(a, a), U1(c, b))) (A^2 -> H(f(c), U1(c, b)),A^2 -> H(f(c), U1(c, b))) (A^2 -> H(f(d), U1(c, b)),A^2 -> H(f(d), U1(c, b))) (A^2 -> H(f(a), U1(e, b)),A^2 -> H(f(a), U1(e, b))) (A^2 -> H(f(a), U1(k, b)),A^2 -> H(f(a), U1(k, b))) (A^2 -> H(f(a), U1(c, c)),A^2 -> H(f(a), U1(c, c))) (A^2 -> H(f(a), U1(c, d)),A^2 -> H(f(a), U1(c, d))) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(d, b)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(d, b)),A^2 -> H(U1(a, a), U1(d, b))) (A^2 -> H(f(c), U1(d, b)),A^2 -> H(f(c), U1(d, b))) (A^2 -> H(f(d), U1(d, b)),A^2 -> H(f(d), U1(d, b))) (A^2 -> H(f(a), U1(k, b)),A^2 -> H(f(a), U1(k, b))) (A^2 -> H(f(a), U1(d, c)),A^2 -> H(f(a), U1(d, c))) (A^2 -> H(f(a), U1(d, d)),A^2 -> H(f(a), U1(d, d))) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(b, c)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(b, c)),A^2 -> H(U1(a, a), U1(b, c))) (A^2 -> H(f(c), U1(b, c)),A^2 -> H(f(c), U1(b, c))) (A^2 -> H(f(d), U1(b, c)),A^2 -> H(f(d), U1(b, c))) (A^2 -> H(f(a), U1(c, c)),A^2 -> H(f(a), U1(c, c))) (A^2 -> H(f(a), U1(d, c)),A^2 -> H(f(a), U1(d, c))) (A^2 -> H(f(a), U1(b, e)),A^2 -> H(f(a), U1(b, e))) (A^2 -> H(f(a), U1(b, k)),A^2 -> H(f(a), U1(b, k))) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(b, d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(b, d)),A^2 -> H(U1(a, a), U1(b, d))) (A^2 -> H(f(c), U1(b, d)),A^2 -> H(f(c), U1(b, d))) (A^2 -> H(f(d), U1(b, d)),A^2 -> H(f(d), U1(b, d))) (A^2 -> H(f(a), U1(c, d)),A^2 -> H(f(a), U1(c, d))) (A^2 -> H(f(a), U1(d, d)),A^2 -> H(f(a), U1(d, d))) (A^2 -> H(f(a), U1(b, k)),A^2 -> H(f(a), U1(b, k))) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, c)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(c, c)),A^2 -> H(U1(a, a), U1(c, c))) (A^2 -> H(f(c), U1(c, c)),A^2 -> H(f(c), U1(c, c))) (A^2 -> H(f(d), U1(c, c)),A^2 -> H(f(d), U1(c, c))) (A^2 -> H(f(a), U1(e, c)),A^2 -> H(f(a), U1(e, c))) (A^2 -> H(f(a), U1(k, c)),A^2 -> H(f(a), U1(k, c))) (A^2 -> H(f(a), U1(c, e)),A^2 -> H(f(a), U1(c, e))) (A^2 -> H(f(a), U1(c, k)),A^2 -> H(f(a), U1(c, k))) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), f(e)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), f(e)),A^2 -> H(U1(a, a), f(e))) (A^2 -> H(f(c), f(e)),A^2 -> H(f(c), f(e))) (A^2 -> H(f(d), f(e)),A^2 -> H(f(d), f(e))) (A^2 -> H(f(a), U1(e, e)),A^2 -> H(f(a), U1(e, e))) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), f(k)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), f(k)),A^2 -> H(U1(a, a), f(k))) (A^2 -> H(f(c), f(k)),A^2 -> H(f(c), f(k))) (A^2 -> H(f(d), f(k)),A^2 -> H(f(d), f(k))) (A^2 -> H(f(a), U1(k, k)),A^2 -> H(f(a), U1(k, k))) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(d, d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(d, d)),A^2 -> H(U1(a, a), U1(d, d))) (A^2 -> H(f(c), U1(d, d)),A^2 -> H(f(c), U1(d, d))) (A^2 -> H(f(d), U1(d, d)),A^2 -> H(f(d), U1(d, d))) (A^2 -> H(f(a), U1(k, d)),A^2 -> H(f(a), U1(k, d))) (A^2 -> H(f(a), U1(d, k)),A^2 -> H(f(a), U1(d, k))) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, a), f(b)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, f(b)),A^2 -> H(a, f(b))) (A^2 -> H(U1(e, c), f(b)),A^2 -> H(U1(e, c), f(b))) (A^2 -> H(U1(e, d), f(b)),A^2 -> H(U1(e, d), f(b))) (A^2 -> H(U1(e, a), U1(b, b)),A^2 -> H(U1(e, a), U1(b, b))) (A^2 -> H(U1(e, a), f(c)),A^2 -> H(U1(e, a), f(c))) (A^2 -> H(U1(e, a), f(d)),A^2 -> H(U1(e, a), f(d))) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, a), f(b)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(b)),A^2 -> H(U1(k, c), f(b))) (A^2 -> H(U1(k, d), f(b)),A^2 -> H(U1(k, d), f(b))) (A^2 -> H(U1(k, a), U1(b, b)),A^2 -> H(U1(k, a), U1(b, b))) (A^2 -> H(U1(k, a), f(c)),A^2 -> H(U1(k, a), f(c))) (A^2 -> H(U1(k, a), f(d)),A^2 -> H(U1(k, a), f(d))) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, d), f(b)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), f(b)),A^2 -> H(U1(e, d), f(b))) (A^2 -> H(U1(k, d), f(b)),A^2 -> H(U1(k, d), f(b))) (A^2 -> H(U1(c, k), f(b)),A^2 -> H(U1(c, k), f(b))) (A^2 -> H(U1(c, d), U1(b, b)),A^2 -> H(U1(c, d), U1(b, b))) (A^2 -> H(U1(c, d), f(c)),A^2 -> H(U1(c, d), f(c))) (A^2 -> H(U1(c, d), f(d)),A^2 -> H(U1(c, d), f(d))) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(b, b)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(b, b)),A^2 -> H(U1(e, a), U1(b, b))) (A^2 -> H(U1(k, a), U1(b, b)),A^2 -> H(U1(k, a), U1(b, b))) (A^2 -> H(U1(c, c), U1(b, b)),A^2 -> H(U1(c, c), U1(b, b))) (A^2 -> H(U1(c, d), U1(b, b)),A^2 -> H(U1(c, d), U1(b, b))) (A^2 -> H(U1(c, a), U1(c, b)),A^2 -> H(U1(c, a), U1(c, b))) (A^2 -> H(U1(c, a), U1(d, b)),A^2 -> H(U1(c, a), U1(d, b))) (A^2 -> H(U1(c, a), U1(b, c)),A^2 -> H(U1(c, a), U1(b, c))) (A^2 -> H(U1(c, a), U1(b, d)),A^2 -> H(U1(c, a), U1(b, d))) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), f(c)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), f(c)),A^2 -> H(U1(e, a), f(c))) (A^2 -> H(U1(k, a), f(c)),A^2 -> H(U1(k, a), f(c))) (A^2 -> H(U1(c, c), f(c)),A^2 -> H(U1(c, c), f(c))) (A^2 -> H(U1(c, d), f(c)),A^2 -> H(U1(c, d), f(c))) (A^2 -> H(U1(c, a), U1(c, c)),A^2 -> H(U1(c, a), U1(c, c))) (A^2 -> H(U1(c, a), f(e)),A^2 -> H(U1(c, a), f(e))) (A^2 -> H(U1(c, a), f(k)),A^2 -> H(U1(c, a), f(k))) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), f(d)),A^2 -> H(U1(e, a), f(d))) (A^2 -> H(U1(k, a), f(d)),A^2 -> H(U1(k, a), f(d))) (A^2 -> H(U1(c, c), f(d)),A^2 -> H(U1(c, c), f(d))) (A^2 -> H(U1(c, d), f(d)),A^2 -> H(U1(c, d), f(d))) (A^2 -> H(U1(c, a), U1(d, d)),A^2 -> H(U1(c, a), U1(d, d))) (A^2 -> H(U1(c, a), f(k)),A^2 -> H(U1(c, a), f(k))) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, c), f(b)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(b)),A^2 -> H(U1(k, c), f(b))) (A^2 -> H(U1(d, e), f(b)),A^2 -> H(U1(d, e), f(b))) (A^2 -> H(U1(d, k), f(b)),A^2 -> H(U1(d, k), f(b))) (A^2 -> H(U1(d, c), U1(b, b)),A^2 -> H(U1(d, c), U1(b, b))) (A^2 -> H(U1(d, c), f(c)),A^2 -> H(U1(d, c), f(c))) (A^2 -> H(U1(d, c), f(d)),A^2 -> H(U1(d, c), f(d))) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(b, b)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(b, b)),A^2 -> H(U1(k, a), U1(b, b))) (A^2 -> H(U1(d, c), U1(b, b)),A^2 -> H(U1(d, c), U1(b, b))) (A^2 -> H(U1(d, d), U1(b, b)),A^2 -> H(U1(d, d), U1(b, b))) (A^2 -> H(U1(d, a), U1(c, b)),A^2 -> H(U1(d, a), U1(c, b))) (A^2 -> H(U1(d, a), U1(d, b)),A^2 -> H(U1(d, a), U1(d, b))) (A^2 -> H(U1(d, a), U1(b, c)),A^2 -> H(U1(d, a), U1(b, c))) (A^2 -> H(U1(d, a), U1(b, d)),A^2 -> H(U1(d, a), U1(b, d))) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), f(c)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), f(c)),A^2 -> H(U1(k, a), f(c))) (A^2 -> H(U1(d, c), f(c)),A^2 -> H(U1(d, c), f(c))) (A^2 -> H(U1(d, d), f(c)),A^2 -> H(U1(d, d), f(c))) (A^2 -> H(U1(d, a), U1(c, c)),A^2 -> H(U1(d, a), U1(c, c))) (A^2 -> H(U1(d, a), f(e)),A^2 -> H(U1(d, a), f(e))) (A^2 -> H(U1(d, a), f(k)),A^2 -> H(U1(d, a), f(k))) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), f(d)),A^2 -> H(U1(k, a), f(d))) (A^2 -> H(U1(d, c), f(d)),A^2 -> H(U1(d, c), f(d))) (A^2 -> H(U1(d, d), f(d)),A^2 -> H(U1(d, d), f(d))) (A^2 -> H(U1(d, a), U1(d, d)),A^2 -> H(U1(d, a), U1(d, d))) (A^2 -> H(U1(d, a), f(k)),A^2 -> H(U1(d, a), f(k))) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, e), f(b)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), f(b)),A^2 -> H(U1(c, e), f(b))) (A^2 -> H(U1(d, e), f(b)),A^2 -> H(U1(d, e), f(b))) (A^2 -> H(U1(a, e), U1(b, b)),A^2 -> H(U1(a, e), U1(b, b))) (A^2 -> H(U1(a, e), f(c)),A^2 -> H(U1(a, e), f(c))) (A^2 -> H(U1(a, e), f(d)),A^2 -> H(U1(a, e), f(d))) ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, k), f(b)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), f(b)),A^2 -> H(U1(c, k), f(b))) (A^2 -> H(U1(d, k), f(b)),A^2 -> H(U1(d, k), f(b))) (A^2 -> H(U1(a, k), U1(b, b)),A^2 -> H(U1(a, k), U1(b, b))) (A^2 -> H(U1(a, k), f(c)),A^2 -> H(U1(a, k), f(c))) (A^2 -> H(U1(a, k), f(d)),A^2 -> H(U1(a, k), f(d))) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(b, b)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, b)),A^2 -> H(U1(c, c), U1(b, b))) (A^2 -> H(U1(d, c), U1(b, b)),A^2 -> H(U1(d, c), U1(b, b))) (A^2 -> H(U1(a, e), U1(b, b)),A^2 -> H(U1(a, e), U1(b, b))) (A^2 -> H(U1(a, k), U1(b, b)),A^2 -> H(U1(a, k), U1(b, b))) (A^2 -> H(U1(a, c), U1(c, b)),A^2 -> H(U1(a, c), U1(c, b))) (A^2 -> H(U1(a, c), U1(d, b)),A^2 -> H(U1(a, c), U1(d, b))) (A^2 -> H(U1(a, c), U1(b, c)),A^2 -> H(U1(a, c), U1(b, c))) (A^2 -> H(U1(a, c), U1(b, d)),A^2 -> H(U1(a, c), U1(b, d))) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), f(c)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(c)),A^2 -> H(U1(c, c), f(c))) (A^2 -> H(U1(d, c), f(c)),A^2 -> H(U1(d, c), f(c))) (A^2 -> H(U1(a, e), f(c)),A^2 -> H(U1(a, e), f(c))) (A^2 -> H(U1(a, k), f(c)),A^2 -> H(U1(a, k), f(c))) (A^2 -> H(U1(a, c), U1(c, c)),A^2 -> H(U1(a, c), U1(c, c))) (A^2 -> H(U1(a, c), f(e)),A^2 -> H(U1(a, c), f(e))) (A^2 -> H(U1(a, c), f(k)),A^2 -> H(U1(a, c), f(k))) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(d)),A^2 -> H(U1(c, c), f(d))) (A^2 -> H(U1(d, c), f(d)),A^2 -> H(U1(d, c), f(d))) (A^2 -> H(U1(a, e), f(d)),A^2 -> H(U1(a, e), f(d))) (A^2 -> H(U1(a, k), f(d)),A^2 -> H(U1(a, k), f(d))) (A^2 -> H(U1(a, c), U1(d, d)),A^2 -> H(U1(a, c), U1(d, d))) (A^2 -> H(U1(a, c), f(k)),A^2 -> H(U1(a, c), f(k))) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(b, b)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(b, b)),A^2 -> H(U1(c, d), U1(b, b))) (A^2 -> H(U1(d, d), U1(b, b)),A^2 -> H(U1(d, d), U1(b, b))) (A^2 -> H(U1(a, k), U1(b, b)),A^2 -> H(U1(a, k), U1(b, b))) (A^2 -> H(U1(a, d), U1(c, b)),A^2 -> H(U1(a, d), U1(c, b))) (A^2 -> H(U1(a, d), U1(d, b)),A^2 -> H(U1(a, d), U1(d, b))) (A^2 -> H(U1(a, d), U1(b, c)),A^2 -> H(U1(a, d), U1(b, c))) (A^2 -> H(U1(a, d), U1(b, d)),A^2 -> H(U1(a, d), U1(b, d))) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), f(c)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), f(c)),A^2 -> H(U1(c, d), f(c))) (A^2 -> H(U1(d, d), f(c)),A^2 -> H(U1(d, d), f(c))) (A^2 -> H(U1(a, k), f(c)),A^2 -> H(U1(a, k), f(c))) (A^2 -> H(U1(a, d), U1(c, c)),A^2 -> H(U1(a, d), U1(c, c))) (A^2 -> H(U1(a, d), f(e)),A^2 -> H(U1(a, d), f(e))) (A^2 -> H(U1(a, d), f(k)),A^2 -> H(U1(a, d), f(k))) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), f(d)),A^2 -> H(U1(c, d), f(d))) (A^2 -> H(U1(d, d), f(d)),A^2 -> H(U1(d, d), f(d))) (A^2 -> H(U1(a, k), f(d)),A^2 -> H(U1(a, k), f(d))) (A^2 -> H(U1(a, d), U1(d, d)),A^2 -> H(U1(a, d), U1(d, d))) (A^2 -> H(U1(a, d), f(k)),A^2 -> H(U1(a, d), f(k))) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(c, b)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(c, b)),A^2 -> H(U1(c, a), U1(c, b))) (A^2 -> H(U1(d, a), U1(c, b)),A^2 -> H(U1(d, a), U1(c, b))) (A^2 -> H(U1(a, c), U1(c, b)),A^2 -> H(U1(a, c), U1(c, b))) (A^2 -> H(U1(a, d), U1(c, b)),A^2 -> H(U1(a, d), U1(c, b))) (A^2 -> H(U1(a, a), U1(e, b)),A^2 -> H(U1(a, a), U1(e, b))) (A^2 -> H(U1(a, a), U1(k, b)),A^2 -> H(U1(a, a), U1(k, b))) (A^2 -> H(U1(a, a), U1(c, c)),A^2 -> H(U1(a, a), U1(c, c))) (A^2 -> H(U1(a, a), U1(c, d)),A^2 -> H(U1(a, a), U1(c, d))) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(d, b)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(d, b)),A^2 -> H(U1(c, a), U1(d, b))) (A^2 -> H(U1(d, a), U1(d, b)),A^2 -> H(U1(d, a), U1(d, b))) (A^2 -> H(U1(a, c), U1(d, b)),A^2 -> H(U1(a, c), U1(d, b))) (A^2 -> H(U1(a, d), U1(d, b)),A^2 -> H(U1(a, d), U1(d, b))) (A^2 -> H(U1(a, a), U1(k, b)),A^2 -> H(U1(a, a), U1(k, b))) (A^2 -> H(U1(a, a), U1(d, c)),A^2 -> H(U1(a, a), U1(d, c))) (A^2 -> H(U1(a, a), U1(d, d)),A^2 -> H(U1(a, a), U1(d, d))) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(b, c)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(b, c)),A^2 -> H(U1(c, a), U1(b, c))) (A^2 -> H(U1(d, a), U1(b, c)),A^2 -> H(U1(d, a), U1(b, c))) (A^2 -> H(U1(a, c), U1(b, c)),A^2 -> H(U1(a, c), U1(b, c))) (A^2 -> H(U1(a, d), U1(b, c)),A^2 -> H(U1(a, d), U1(b, c))) (A^2 -> H(U1(a, a), U1(c, c)),A^2 -> H(U1(a, a), U1(c, c))) (A^2 -> H(U1(a, a), U1(d, c)),A^2 -> H(U1(a, a), U1(d, c))) (A^2 -> H(U1(a, a), U1(b, e)),A^2 -> H(U1(a, a), U1(b, e))) (A^2 -> H(U1(a, a), U1(b, k)),A^2 -> H(U1(a, a), U1(b, k))) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(b, d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(b, d)),A^2 -> H(U1(c, a), U1(b, d))) (A^2 -> H(U1(d, a), U1(b, d)),A^2 -> H(U1(d, a), U1(b, d))) (A^2 -> H(U1(a, c), U1(b, d)),A^2 -> H(U1(a, c), U1(b, d))) (A^2 -> H(U1(a, d), U1(b, d)),A^2 -> H(U1(a, d), U1(b, d))) (A^2 -> H(U1(a, a), U1(c, d)),A^2 -> H(U1(a, a), U1(c, d))) (A^2 -> H(U1(a, a), U1(d, d)),A^2 -> H(U1(a, a), U1(d, d))) (A^2 -> H(U1(a, a), U1(b, k)),A^2 -> H(U1(a, a), U1(b, k))) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(c, c)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(c, c)),A^2 -> H(U1(c, a), U1(c, c))) (A^2 -> H(U1(d, a), U1(c, c)),A^2 -> H(U1(d, a), U1(c, c))) (A^2 -> H(U1(a, c), U1(c, c)),A^2 -> H(U1(a, c), U1(c, c))) (A^2 -> H(U1(a, d), U1(c, c)),A^2 -> H(U1(a, d), U1(c, c))) (A^2 -> H(U1(a, a), U1(e, c)),A^2 -> H(U1(a, a), U1(e, c))) (A^2 -> H(U1(a, a), U1(k, c)),A^2 -> H(U1(a, a), U1(k, c))) (A^2 -> H(U1(a, a), U1(c, e)),A^2 -> H(U1(a, a), U1(c, e))) (A^2 -> H(U1(a, a), U1(c, k)),A^2 -> H(U1(a, a), U1(c, k))) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), f(e)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), f(e)),A^2 -> H(U1(c, a), f(e))) (A^2 -> H(U1(d, a), f(e)),A^2 -> H(U1(d, a), f(e))) (A^2 -> H(U1(a, c), f(e)),A^2 -> H(U1(a, c), f(e))) (A^2 -> H(U1(a, d), f(e)),A^2 -> H(U1(a, d), f(e))) (A^2 -> H(U1(a, a), U1(e, e)),A^2 -> H(U1(a, a), U1(e, e))) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), f(k)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), f(k)),A^2 -> H(U1(c, a), f(k))) (A^2 -> H(U1(d, a), f(k)),A^2 -> H(U1(d, a), f(k))) (A^2 -> H(U1(a, c), f(k)),A^2 -> H(U1(a, c), f(k))) (A^2 -> H(U1(a, d), f(k)),A^2 -> H(U1(a, d), f(k))) (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) ---------------------------------------- (118) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(d, d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (119) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(d, d)),A^2 -> H(U1(c, a), U1(d, d))) (A^2 -> H(U1(d, a), U1(d, d)),A^2 -> H(U1(d, a), U1(d, d))) (A^2 -> H(U1(a, c), U1(d, d)),A^2 -> H(U1(a, c), U1(d, d))) (A^2 -> H(U1(a, d), U1(d, d)),A^2 -> H(U1(a, d), U1(d, d))) (A^2 -> H(U1(a, a), U1(k, d)),A^2 -> H(U1(a, a), U1(k, d))) (A^2 -> H(U1(a, a), U1(d, k)),A^2 -> H(U1(a, a), U1(d, k))) ---------------------------------------- (120) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, c), f(b)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (121) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(b)),A^2 -> H(c, f(b))) (A^2 -> H(U1(e, e), f(b)),A^2 -> H(U1(e, e), f(b))) (A^2 -> H(U1(e, k), f(b)),A^2 -> H(U1(e, k), f(b))) (A^2 -> H(U1(e, c), U1(b, b)),A^2 -> H(U1(e, c), U1(b, b))) (A^2 -> H(U1(e, c), f(c)),A^2 -> H(U1(e, c), f(c))) (A^2 -> H(U1(e, c), f(d)),A^2 -> H(U1(e, c), f(d))) ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, c), f(b)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(b)),A^2 -> H(U1(k, e), f(b))) (A^2 -> H(U1(k, k), f(b)),A^2 -> H(U1(k, k), f(b))) (A^2 -> H(U1(k, c), U1(b, b)),A^2 -> H(U1(k, c), U1(b, b))) (A^2 -> H(U1(k, c), f(c)),A^2 -> H(U1(k, c), f(c))) (A^2 -> H(U1(k, c), f(d)),A^2 -> H(U1(k, c), f(d))) ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, e), f(b)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(b)),A^2 -> H(U1(e, e), f(b))) (A^2 -> H(U1(k, e), f(b)),A^2 -> H(U1(k, e), f(b))) (A^2 -> H(U1(c, e), U1(b, b)),A^2 -> H(U1(c, e), U1(b, b))) (A^2 -> H(U1(c, e), f(c)),A^2 -> H(U1(c, e), f(c))) (A^2 -> H(U1(c, e), f(d)),A^2 -> H(U1(c, e), f(d))) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, k), f(b)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), f(b)),A^2 -> H(U1(e, k), f(b))) (A^2 -> H(U1(k, k), f(b)),A^2 -> H(U1(k, k), f(b))) (A^2 -> H(U1(c, k), U1(b, b)),A^2 -> H(U1(c, k), U1(b, b))) (A^2 -> H(U1(c, k), f(c)),A^2 -> H(U1(c, k), f(c))) (A^2 -> H(U1(c, k), f(d)),A^2 -> H(U1(c, k), f(d))) ---------------------------------------- (128) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(b, b)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (129) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(b, b)),A^2 -> H(U1(e, c), U1(b, b))) (A^2 -> H(U1(k, c), U1(b, b)),A^2 -> H(U1(k, c), U1(b, b))) (A^2 -> H(U1(c, e), U1(b, b)),A^2 -> H(U1(c, e), U1(b, b))) (A^2 -> H(U1(c, k), U1(b, b)),A^2 -> H(U1(c, k), U1(b, b))) (A^2 -> H(U1(c, c), U1(c, b)),A^2 -> H(U1(c, c), U1(c, b))) (A^2 -> H(U1(c, c), U1(d, b)),A^2 -> H(U1(c, c), U1(d, b))) (A^2 -> H(U1(c, c), U1(b, c)),A^2 -> H(U1(c, c), U1(b, c))) (A^2 -> H(U1(c, c), U1(b, d)),A^2 -> H(U1(c, c), U1(b, d))) ---------------------------------------- (130) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), f(c)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (131) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), f(c)),A^2 -> H(U1(e, c), f(c))) (A^2 -> H(U1(k, c), f(c)),A^2 -> H(U1(k, c), f(c))) (A^2 -> H(U1(c, e), f(c)),A^2 -> H(U1(c, e), f(c))) (A^2 -> H(U1(c, k), f(c)),A^2 -> H(U1(c, k), f(c))) (A^2 -> H(U1(c, c), U1(c, c)),A^2 -> H(U1(c, c), U1(c, c))) (A^2 -> H(U1(c, c), f(e)),A^2 -> H(U1(c, c), f(e))) (A^2 -> H(U1(c, c), f(k)),A^2 -> H(U1(c, c), f(k))) ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), f(d)),A^2 -> H(U1(e, c), f(d))) (A^2 -> H(U1(k, c), f(d)),A^2 -> H(U1(k, c), f(d))) (A^2 -> H(U1(c, e), f(d)),A^2 -> H(U1(c, e), f(d))) (A^2 -> H(U1(c, k), f(d)),A^2 -> H(U1(c, k), f(d))) (A^2 -> H(U1(c, c), U1(d, d)),A^2 -> H(U1(c, c), U1(d, d))) (A^2 -> H(U1(c, c), f(k)),A^2 -> H(U1(c, c), f(k))) ---------------------------------------- (134) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, e), f(b)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (135) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(b)),A^2 -> H(e, f(b))) (A^2 -> H(U1(e, e), U1(b, b)),A^2 -> H(U1(e, e), U1(b, b))) (A^2 -> H(U1(e, e), f(c)),A^2 -> H(U1(e, e), f(c))) (A^2 -> H(U1(e, e), f(d)),A^2 -> H(U1(e, e), f(d))) ---------------------------------------- (136) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(e), U1(b, b)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (137) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, b)),A^2 -> H(U1(e, e), U1(b, b))) (A^2 -> H(f(e), U1(c, b)),A^2 -> H(f(e), U1(c, b))) (A^2 -> H(f(e), U1(d, b)),A^2 -> H(f(e), U1(d, b))) (A^2 -> H(f(e), U1(b, c)),A^2 -> H(f(e), U1(b, c))) (A^2 -> H(f(e), U1(b, d)),A^2 -> H(f(e), U1(b, d))) ---------------------------------------- (138) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(e), f(c)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (139) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(c)),A^2 -> H(U1(e, e), f(c))) (A^2 -> H(f(e), U1(c, c)),A^2 -> H(f(e), U1(c, c))) (A^2 -> H(f(e), f(e)),A^2 -> H(f(e), f(e))) (A^2 -> H(f(e), f(k)),A^2 -> H(f(e), f(k))) ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(e), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(d)),A^2 -> H(U1(e, e), f(d))) (A^2 -> H(f(e), U1(d, d)),A^2 -> H(f(e), U1(d, d))) (A^2 -> H(f(e), f(k)),A^2 -> H(f(e), f(k))) ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(k), U1(b, b)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(f(k), U1(c, b)),A^2 -> H(f(k), U1(c, b))) (A^2 -> H(f(k), U1(d, b)),A^2 -> H(f(k), U1(d, b))) (A^2 -> H(f(k), U1(b, c)),A^2 -> H(f(k), U1(b, c))) (A^2 -> H(f(k), U1(b, d)),A^2 -> H(f(k), U1(b, d))) ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(k), f(c)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(f(k), U1(c, c)),A^2 -> H(f(k), U1(c, c))) (A^2 -> H(f(k), f(e)),A^2 -> H(f(k), f(e))) (A^2 -> H(f(k), f(k)),A^2 -> H(f(k), f(k))) ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(k), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) (A^2 -> H(f(k), U1(d, d)),A^2 -> H(f(k), U1(d, d))) (A^2 -> H(f(k), f(k)),A^2 -> H(f(k), f(k))) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), U1(c, b)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, b)),A^2 -> H(U1(c, c), U1(c, b))) (A^2 -> H(f(e), U1(c, b)),A^2 -> H(f(e), U1(c, b))) (A^2 -> H(f(k), U1(c, b)),A^2 -> H(f(k), U1(c, b))) (A^2 -> H(f(c), U1(e, b)),A^2 -> H(f(c), U1(e, b))) (A^2 -> H(f(c), U1(k, b)),A^2 -> H(f(c), U1(k, b))) (A^2 -> H(f(c), U1(c, c)),A^2 -> H(f(c), U1(c, c))) (A^2 -> H(f(c), U1(c, d)),A^2 -> H(f(c), U1(c, d))) ---------------------------------------- (152) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), U1(d, b)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, b)),A^2 -> H(U1(c, c), U1(d, b))) (A^2 -> H(f(e), U1(d, b)),A^2 -> H(f(e), U1(d, b))) (A^2 -> H(f(k), U1(d, b)),A^2 -> H(f(k), U1(d, b))) (A^2 -> H(f(c), U1(k, b)),A^2 -> H(f(c), U1(k, b))) (A^2 -> H(f(c), U1(d, c)),A^2 -> H(f(c), U1(d, c))) (A^2 -> H(f(c), U1(d, d)),A^2 -> H(f(c), U1(d, d))) ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), U1(b, c)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, c)),A^2 -> H(U1(c, c), U1(b, c))) (A^2 -> H(f(e), U1(b, c)),A^2 -> H(f(e), U1(b, c))) (A^2 -> H(f(k), U1(b, c)),A^2 -> H(f(k), U1(b, c))) (A^2 -> H(f(c), U1(c, c)),A^2 -> H(f(c), U1(c, c))) (A^2 -> H(f(c), U1(d, c)),A^2 -> H(f(c), U1(d, c))) (A^2 -> H(f(c), U1(b, e)),A^2 -> H(f(c), U1(b, e))) (A^2 -> H(f(c), U1(b, k)),A^2 -> H(f(c), U1(b, k))) ---------------------------------------- (156) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), U1(b, d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (157) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, d)),A^2 -> H(U1(c, c), U1(b, d))) (A^2 -> H(f(e), U1(b, d)),A^2 -> H(f(e), U1(b, d))) (A^2 -> H(f(k), U1(b, d)),A^2 -> H(f(k), U1(b, d))) (A^2 -> H(f(c), U1(c, d)),A^2 -> H(f(c), U1(c, d))) (A^2 -> H(f(c), U1(d, d)),A^2 -> H(f(c), U1(d, d))) (A^2 -> H(f(c), U1(b, k)),A^2 -> H(f(c), U1(b, k))) ---------------------------------------- (158) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), U1(d, d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (159) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, d)),A^2 -> H(U1(c, c), U1(d, d))) (A^2 -> H(f(e), U1(d, d)),A^2 -> H(f(e), U1(d, d))) (A^2 -> H(f(k), U1(d, d)),A^2 -> H(f(k), U1(d, d))) (A^2 -> H(f(c), U1(k, d)),A^2 -> H(f(c), U1(k, d))) (A^2 -> H(f(c), U1(d, k)),A^2 -> H(f(c), U1(d, k))) ---------------------------------------- (160) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), f(k)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (161) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(k)),A^2 -> H(U1(c, c), f(k))) (A^2 -> H(f(e), f(k)),A^2 -> H(f(e), f(k))) (A^2 -> H(f(k), f(k)),A^2 -> H(f(k), f(k))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) ---------------------------------------- (162) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, d), f(b)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (163) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(b)),A^2 -> H(U1(k, k), f(b))) (A^2 -> H(U1(k, d), U1(b, b)),A^2 -> H(U1(k, d), U1(b, b))) (A^2 -> H(U1(k, d), f(c)),A^2 -> H(U1(k, d), f(c))) (A^2 -> H(U1(k, d), f(d)),A^2 -> H(U1(k, d), f(d))) ---------------------------------------- (164) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, k), f(b)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (165) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(b)),A^2 -> H(U1(k, k), f(b))) (A^2 -> H(U1(d, k), U1(b, b)),A^2 -> H(U1(d, k), U1(b, b))) (A^2 -> H(U1(d, k), f(c)),A^2 -> H(U1(d, k), f(c))) (A^2 -> H(U1(d, k), f(d)),A^2 -> H(U1(d, k), f(d))) ---------------------------------------- (166) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, d), U1(b, b)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (167) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(b, b)),A^2 -> H(U1(k, d), U1(b, b))) (A^2 -> H(U1(d, k), U1(b, b)),A^2 -> H(U1(d, k), U1(b, b))) (A^2 -> H(U1(d, d), U1(c, b)),A^2 -> H(U1(d, d), U1(c, b))) (A^2 -> H(U1(d, d), U1(d, b)),A^2 -> H(U1(d, d), U1(d, b))) (A^2 -> H(U1(d, d), U1(b, c)),A^2 -> H(U1(d, d), U1(b, c))) (A^2 -> H(U1(d, d), U1(b, d)),A^2 -> H(U1(d, d), U1(b, d))) ---------------------------------------- (168) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, d), f(c)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (169) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), f(c)),A^2 -> H(U1(k, d), f(c))) (A^2 -> H(U1(d, k), f(c)),A^2 -> H(U1(d, k), f(c))) (A^2 -> H(U1(d, d), U1(c, c)),A^2 -> H(U1(d, d), U1(c, c))) (A^2 -> H(U1(d, d), f(e)),A^2 -> H(U1(d, d), f(e))) (A^2 -> H(U1(d, d), f(k)),A^2 -> H(U1(d, d), f(k))) ---------------------------------------- (170) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (171) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), f(d)),A^2 -> H(U1(k, d), f(d))) (A^2 -> H(U1(d, k), f(d)),A^2 -> H(U1(d, k), f(d))) (A^2 -> H(U1(d, d), U1(d, d)),A^2 -> H(U1(d, d), U1(d, d))) (A^2 -> H(U1(d, d), f(k)),A^2 -> H(U1(d, d), f(k))) ---------------------------------------- (172) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), U1(c, b)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (173) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(c, b)),A^2 -> H(U1(d, d), U1(c, b))) (A^2 -> H(f(k), U1(c, b)),A^2 -> H(f(k), U1(c, b))) (A^2 -> H(f(d), U1(e, b)),A^2 -> H(f(d), U1(e, b))) (A^2 -> H(f(d), U1(k, b)),A^2 -> H(f(d), U1(k, b))) (A^2 -> H(f(d), U1(c, c)),A^2 -> H(f(d), U1(c, c))) (A^2 -> H(f(d), U1(c, d)),A^2 -> H(f(d), U1(c, d))) ---------------------------------------- (174) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), U1(d, b)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (175) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(d, b)),A^2 -> H(U1(d, d), U1(d, b))) (A^2 -> H(f(k), U1(d, b)),A^2 -> H(f(k), U1(d, b))) (A^2 -> H(f(d), U1(k, b)),A^2 -> H(f(d), U1(k, b))) (A^2 -> H(f(d), U1(d, c)),A^2 -> H(f(d), U1(d, c))) (A^2 -> H(f(d), U1(d, d)),A^2 -> H(f(d), U1(d, d))) ---------------------------------------- (176) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), U1(b, c)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (177) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(b, c)),A^2 -> H(U1(d, d), U1(b, c))) (A^2 -> H(f(k), U1(b, c)),A^2 -> H(f(k), U1(b, c))) (A^2 -> H(f(d), U1(c, c)),A^2 -> H(f(d), U1(c, c))) (A^2 -> H(f(d), U1(d, c)),A^2 -> H(f(d), U1(d, c))) (A^2 -> H(f(d), U1(b, e)),A^2 -> H(f(d), U1(b, e))) (A^2 -> H(f(d), U1(b, k)),A^2 -> H(f(d), U1(b, k))) ---------------------------------------- (178) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), U1(b, d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (179) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(b, d)),A^2 -> H(U1(d, d), U1(b, d))) (A^2 -> H(f(k), U1(b, d)),A^2 -> H(f(k), U1(b, d))) (A^2 -> H(f(d), U1(c, d)),A^2 -> H(f(d), U1(c, d))) (A^2 -> H(f(d), U1(d, d)),A^2 -> H(f(d), U1(d, d))) (A^2 -> H(f(d), U1(b, k)),A^2 -> H(f(d), U1(b, k))) ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), U1(c, c)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (181) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(c, c)),A^2 -> H(U1(d, d), U1(c, c))) (A^2 -> H(f(k), U1(c, c)),A^2 -> H(f(k), U1(c, c))) (A^2 -> H(f(d), U1(e, c)),A^2 -> H(f(d), U1(e, c))) (A^2 -> H(f(d), U1(k, c)),A^2 -> H(f(d), U1(k, c))) (A^2 -> H(f(d), U1(c, e)),A^2 -> H(f(d), U1(c, e))) (A^2 -> H(f(d), U1(c, k)),A^2 -> H(f(d), U1(c, k))) ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), f(e)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), f(e)),A^2 -> H(U1(d, d), f(e))) (A^2 -> H(f(k), f(e)),A^2 -> H(f(k), f(e))) (A^2 -> H(f(d), U1(e, e)),A^2 -> H(f(d), U1(e, e))) ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), f(k)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (185) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), f(k)),A^2 -> H(U1(d, d), f(k))) (A^2 -> H(f(k), f(k)),A^2 -> H(f(k), f(k))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (186) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(e, b)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (187) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(e, b)),A^2 -> H(U1(a, a), U1(e, b))) (A^2 -> H(f(c), U1(e, b)),A^2 -> H(f(c), U1(e, b))) (A^2 -> H(f(d), U1(e, b)),A^2 -> H(f(d), U1(e, b))) (A^2 -> H(f(a), b),A^2 -> H(f(a), b)) (A^2 -> H(f(a), U1(e, c)),A^2 -> H(f(a), U1(e, c))) (A^2 -> H(f(a), U1(e, d)),A^2 -> H(f(a), U1(e, d))) ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(k, b)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (189) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(k, b)),A^2 -> H(U1(a, a), U1(k, b))) (A^2 -> H(f(c), U1(k, b)),A^2 -> H(f(c), U1(k, b))) (A^2 -> H(f(d), U1(k, b)),A^2 -> H(f(d), U1(k, b))) (A^2 -> H(f(a), U1(k, c)),A^2 -> H(f(a), U1(k, c))) (A^2 -> H(f(a), U1(k, d)),A^2 -> H(f(a), U1(k, d))) ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (191) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(c, d)),A^2 -> H(U1(a, a), U1(c, d))) (A^2 -> H(f(c), U1(c, d)),A^2 -> H(f(c), U1(c, d))) (A^2 -> H(f(d), U1(c, d)),A^2 -> H(f(d), U1(c, d))) (A^2 -> H(f(a), U1(e, d)),A^2 -> H(f(a), U1(e, d))) (A^2 -> H(f(a), U1(k, d)),A^2 -> H(f(a), U1(k, d))) (A^2 -> H(f(a), U1(c, k)),A^2 -> H(f(a), U1(c, k))) ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(d, c)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (193) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(d, c)),A^2 -> H(U1(a, a), U1(d, c))) (A^2 -> H(f(c), U1(d, c)),A^2 -> H(f(c), U1(d, c))) (A^2 -> H(f(d), U1(d, c)),A^2 -> H(f(d), U1(d, c))) (A^2 -> H(f(a), U1(k, c)),A^2 -> H(f(a), U1(k, c))) (A^2 -> H(f(a), U1(d, e)),A^2 -> H(f(a), U1(d, e))) (A^2 -> H(f(a), U1(d, k)),A^2 -> H(f(a), U1(d, k))) ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(b, e)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (195) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(b, e)),A^2 -> H(U1(a, a), U1(b, e))) (A^2 -> H(f(c), U1(b, e)),A^2 -> H(f(c), U1(b, e))) (A^2 -> H(f(d), U1(b, e)),A^2 -> H(f(d), U1(b, e))) (A^2 -> H(f(a), U1(c, e)),A^2 -> H(f(a), U1(c, e))) (A^2 -> H(f(a), U1(d, e)),A^2 -> H(f(a), U1(d, e))) ---------------------------------------- (196) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(b, k)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (197) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(b, k)),A^2 -> H(U1(a, a), U1(b, k))) (A^2 -> H(f(c), U1(b, k)),A^2 -> H(f(c), U1(b, k))) (A^2 -> H(f(d), U1(b, k)),A^2 -> H(f(d), U1(b, k))) (A^2 -> H(f(a), U1(c, k)),A^2 -> H(f(a), U1(c, k))) (A^2 -> H(f(a), U1(d, k)),A^2 -> H(f(a), U1(d, k))) ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), U1(c, c)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (199) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, c)),A^2 -> H(U1(c, c), U1(c, c))) (A^2 -> H(f(e), U1(c, c)),A^2 -> H(f(e), U1(c, c))) (A^2 -> H(f(k), U1(c, c)),A^2 -> H(f(k), U1(c, c))) (A^2 -> H(f(c), U1(e, c)),A^2 -> H(f(c), U1(e, c))) (A^2 -> H(f(c), U1(k, c)),A^2 -> H(f(c), U1(k, c))) (A^2 -> H(f(c), U1(c, e)),A^2 -> H(f(c), U1(c, e))) (A^2 -> H(f(c), U1(c, k)),A^2 -> H(f(c), U1(c, k))) ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(e, c)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (201) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(e, c)),A^2 -> H(U1(a, a), U1(e, c))) (A^2 -> H(f(c), U1(e, c)),A^2 -> H(f(c), U1(e, c))) (A^2 -> H(f(d), U1(e, c)),A^2 -> H(f(d), U1(e, c))) (A^2 -> H(f(a), c),A^2 -> H(f(a), c)) (A^2 -> H(f(a), U1(e, e)),A^2 -> H(f(a), U1(e, e))) (A^2 -> H(f(a), U1(e, k)),A^2 -> H(f(a), U1(e, k))) ---------------------------------------- (202) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(k, c)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (203) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(k, c)),A^2 -> H(U1(a, a), U1(k, c))) (A^2 -> H(f(c), U1(k, c)),A^2 -> H(f(c), U1(k, c))) (A^2 -> H(f(d), U1(k, c)),A^2 -> H(f(d), U1(k, c))) (A^2 -> H(f(a), U1(k, e)),A^2 -> H(f(a), U1(k, e))) (A^2 -> H(f(a), U1(k, k)),A^2 -> H(f(a), U1(k, k))) ---------------------------------------- (204) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, e)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (205) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(c, e)),A^2 -> H(U1(a, a), U1(c, e))) (A^2 -> H(f(c), U1(c, e)),A^2 -> H(f(c), U1(c, e))) (A^2 -> H(f(d), U1(c, e)),A^2 -> H(f(d), U1(c, e))) (A^2 -> H(f(a), U1(e, e)),A^2 -> H(f(a), U1(e, e))) (A^2 -> H(f(a), U1(k, e)),A^2 -> H(f(a), U1(k, e))) ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(c, k)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (207) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(c, k)),A^2 -> H(U1(a, a), U1(c, k))) (A^2 -> H(f(c), U1(c, k)),A^2 -> H(f(c), U1(c, k))) (A^2 -> H(f(d), U1(c, k)),A^2 -> H(f(d), U1(c, k))) (A^2 -> H(f(a), U1(e, k)),A^2 -> H(f(a), U1(e, k))) (A^2 -> H(f(a), U1(k, k)),A^2 -> H(f(a), U1(k, k))) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(c), f(e)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(e)),A^2 -> H(U1(c, c), f(e))) (A^2 -> H(f(e), f(e)),A^2 -> H(f(e), f(e))) (A^2 -> H(f(k), f(e)),A^2 -> H(f(k), f(e))) (A^2 -> H(f(c), U1(e, e)),A^2 -> H(f(c), U1(e, e))) ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(e, e)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (211) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(e, e)),A^2 -> H(U1(a, a), U1(e, e))) (A^2 -> H(f(c), U1(e, e)),A^2 -> H(f(c), U1(e, e))) (A^2 -> H(f(d), U1(e, e)),A^2 -> H(f(d), U1(e, e))) (A^2 -> H(f(a), e),A^2 -> H(f(a), e)) ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (213) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(d), U1(d, d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (215) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(d, d)),A^2 -> H(U1(d, d), U1(d, d))) (A^2 -> H(f(k), U1(d, d)),A^2 -> H(f(k), U1(d, d))) (A^2 -> H(f(d), U1(k, d)),A^2 -> H(f(d), U1(k, d))) (A^2 -> H(f(d), U1(d, k)),A^2 -> H(f(d), U1(d, k))) ---------------------------------------- (216) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(k, d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (217) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(k, d)),A^2 -> H(U1(a, a), U1(k, d))) (A^2 -> H(f(c), U1(k, d)),A^2 -> H(f(c), U1(k, d))) (A^2 -> H(f(d), U1(k, d)),A^2 -> H(f(d), U1(k, d))) (A^2 -> H(f(a), U1(k, k)),A^2 -> H(f(a), U1(k, k))) ---------------------------------------- (218) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(f(a), U1(d, k)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (219) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(d, k)),A^2 -> H(U1(a, a), U1(d, k))) (A^2 -> H(f(c), U1(d, k)),A^2 -> H(f(c), U1(d, k))) (A^2 -> H(f(d), U1(d, k)),A^2 -> H(f(d), U1(d, k))) (A^2 -> H(f(a), U1(k, k)),A^2 -> H(f(a), U1(k, k))) ---------------------------------------- (220) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(a, f(b)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (221) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(b)),A^2 -> H(c, f(b))) (A^2 -> H(d, f(b)),A^2 -> H(d, f(b))) (A^2 -> H(a, U1(b, b)),A^2 -> H(a, U1(b, b))) (A^2 -> H(a, f(c)),A^2 -> H(a, f(c))) (A^2 -> H(a, f(d)),A^2 -> H(a, f(d))) ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, d), f(b)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (223) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, f(b)),A^2 -> H(d, f(b))) (A^2 -> H(U1(e, k), f(b)),A^2 -> H(U1(e, k), f(b))) (A^2 -> H(U1(e, d), U1(b, b)),A^2 -> H(U1(e, d), U1(b, b))) (A^2 -> H(U1(e, d), f(c)),A^2 -> H(U1(e, d), f(c))) (A^2 -> H(U1(e, d), f(d)),A^2 -> H(U1(e, d), f(d))) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, a), U1(b, b)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (225) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(b, b)),A^2 -> H(a, U1(b, b))) (A^2 -> H(U1(e, c), U1(b, b)),A^2 -> H(U1(e, c), U1(b, b))) (A^2 -> H(U1(e, d), U1(b, b)),A^2 -> H(U1(e, d), U1(b, b))) (A^2 -> H(U1(e, a), U1(c, b)),A^2 -> H(U1(e, a), U1(c, b))) (A^2 -> H(U1(e, a), U1(d, b)),A^2 -> H(U1(e, a), U1(d, b))) (A^2 -> H(U1(e, a), U1(b, c)),A^2 -> H(U1(e, a), U1(b, c))) (A^2 -> H(U1(e, a), U1(b, d)),A^2 -> H(U1(e, a), U1(b, d))) ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, a), f(c)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, f(c)),A^2 -> H(a, f(c))) (A^2 -> H(U1(e, c), f(c)),A^2 -> H(U1(e, c), f(c))) (A^2 -> H(U1(e, d), f(c)),A^2 -> H(U1(e, d), f(c))) (A^2 -> H(U1(e, a), U1(c, c)),A^2 -> H(U1(e, a), U1(c, c))) (A^2 -> H(U1(e, a), f(e)),A^2 -> H(U1(e, a), f(e))) (A^2 -> H(U1(e, a), f(k)),A^2 -> H(U1(e, a), f(k))) ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, a), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (229) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, f(d)),A^2 -> H(a, f(d))) (A^2 -> H(U1(e, c), f(d)),A^2 -> H(U1(e, c), f(d))) (A^2 -> H(U1(e, d), f(d)),A^2 -> H(U1(e, d), f(d))) (A^2 -> H(U1(e, a), U1(d, d)),A^2 -> H(U1(e, a), U1(d, d))) (A^2 -> H(U1(e, a), f(k)),A^2 -> H(U1(e, a), f(k))) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, a), U1(b, b)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, b)),A^2 -> H(U1(k, c), U1(b, b))) (A^2 -> H(U1(k, d), U1(b, b)),A^2 -> H(U1(k, d), U1(b, b))) (A^2 -> H(U1(k, a), U1(c, b)),A^2 -> H(U1(k, a), U1(c, b))) (A^2 -> H(U1(k, a), U1(d, b)),A^2 -> H(U1(k, a), U1(d, b))) (A^2 -> H(U1(k, a), U1(b, c)),A^2 -> H(U1(k, a), U1(b, c))) (A^2 -> H(U1(k, a), U1(b, d)),A^2 -> H(U1(k, a), U1(b, d))) ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, a), f(c)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(c)),A^2 -> H(U1(k, c), f(c))) (A^2 -> H(U1(k, d), f(c)),A^2 -> H(U1(k, d), f(c))) (A^2 -> H(U1(k, a), U1(c, c)),A^2 -> H(U1(k, a), U1(c, c))) (A^2 -> H(U1(k, a), f(e)),A^2 -> H(U1(k, a), f(e))) (A^2 -> H(U1(k, a), f(k)),A^2 -> H(U1(k, a), f(k))) ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, a), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(d)),A^2 -> H(U1(k, c), f(d))) (A^2 -> H(U1(k, d), f(d)),A^2 -> H(U1(k, d), f(d))) (A^2 -> H(U1(k, a), U1(d, d)),A^2 -> H(U1(k, a), U1(d, d))) (A^2 -> H(U1(k, a), f(k)),A^2 -> H(U1(k, a), f(k))) ---------------------------------------- (236) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, d), U1(b, b)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (237) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(b, b)),A^2 -> H(U1(e, d), U1(b, b))) (A^2 -> H(U1(k, d), U1(b, b)),A^2 -> H(U1(k, d), U1(b, b))) (A^2 -> H(U1(c, k), U1(b, b)),A^2 -> H(U1(c, k), U1(b, b))) (A^2 -> H(U1(c, d), U1(c, b)),A^2 -> H(U1(c, d), U1(c, b))) (A^2 -> H(U1(c, d), U1(d, b)),A^2 -> H(U1(c, d), U1(d, b))) (A^2 -> H(U1(c, d), U1(b, c)),A^2 -> H(U1(c, d), U1(b, c))) (A^2 -> H(U1(c, d), U1(b, d)),A^2 -> H(U1(c, d), U1(b, d))) ---------------------------------------- (238) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, d), f(c)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (239) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), f(c)),A^2 -> H(U1(e, d), f(c))) (A^2 -> H(U1(k, d), f(c)),A^2 -> H(U1(k, d), f(c))) (A^2 -> H(U1(c, k), f(c)),A^2 -> H(U1(c, k), f(c))) (A^2 -> H(U1(c, d), U1(c, c)),A^2 -> H(U1(c, d), U1(c, c))) (A^2 -> H(U1(c, d), f(e)),A^2 -> H(U1(c, d), f(e))) (A^2 -> H(U1(c, d), f(k)),A^2 -> H(U1(c, d), f(k))) ---------------------------------------- (240) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (241) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), f(d)),A^2 -> H(U1(e, d), f(d))) (A^2 -> H(U1(k, d), f(d)),A^2 -> H(U1(k, d), f(d))) (A^2 -> H(U1(c, k), f(d)),A^2 -> H(U1(c, k), f(d))) (A^2 -> H(U1(c, d), U1(d, d)),A^2 -> H(U1(c, d), U1(d, d))) (A^2 -> H(U1(c, d), f(k)),A^2 -> H(U1(c, d), f(k))) ---------------------------------------- (242) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(c, b)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (243) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(c, b)),A^2 -> H(U1(e, a), U1(c, b))) (A^2 -> H(U1(k, a), U1(c, b)),A^2 -> H(U1(k, a), U1(c, b))) (A^2 -> H(U1(c, c), U1(c, b)),A^2 -> H(U1(c, c), U1(c, b))) (A^2 -> H(U1(c, d), U1(c, b)),A^2 -> H(U1(c, d), U1(c, b))) (A^2 -> H(U1(c, a), U1(e, b)),A^2 -> H(U1(c, a), U1(e, b))) (A^2 -> H(U1(c, a), U1(k, b)),A^2 -> H(U1(c, a), U1(k, b))) (A^2 -> H(U1(c, a), U1(c, c)),A^2 -> H(U1(c, a), U1(c, c))) (A^2 -> H(U1(c, a), U1(c, d)),A^2 -> H(U1(c, a), U1(c, d))) ---------------------------------------- (244) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(d, b)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (245) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(d, b)),A^2 -> H(U1(e, a), U1(d, b))) (A^2 -> H(U1(k, a), U1(d, b)),A^2 -> H(U1(k, a), U1(d, b))) (A^2 -> H(U1(c, c), U1(d, b)),A^2 -> H(U1(c, c), U1(d, b))) (A^2 -> H(U1(c, d), U1(d, b)),A^2 -> H(U1(c, d), U1(d, b))) (A^2 -> H(U1(c, a), U1(k, b)),A^2 -> H(U1(c, a), U1(k, b))) (A^2 -> H(U1(c, a), U1(d, c)),A^2 -> H(U1(c, a), U1(d, c))) (A^2 -> H(U1(c, a), U1(d, d)),A^2 -> H(U1(c, a), U1(d, d))) ---------------------------------------- (246) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(b, c)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (247) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(b, c)),A^2 -> H(U1(e, a), U1(b, c))) (A^2 -> H(U1(k, a), U1(b, c)),A^2 -> H(U1(k, a), U1(b, c))) (A^2 -> H(U1(c, c), U1(b, c)),A^2 -> H(U1(c, c), U1(b, c))) (A^2 -> H(U1(c, d), U1(b, c)),A^2 -> H(U1(c, d), U1(b, c))) (A^2 -> H(U1(c, a), U1(c, c)),A^2 -> H(U1(c, a), U1(c, c))) (A^2 -> H(U1(c, a), U1(d, c)),A^2 -> H(U1(c, a), U1(d, c))) (A^2 -> H(U1(c, a), U1(b, e)),A^2 -> H(U1(c, a), U1(b, e))) (A^2 -> H(U1(c, a), U1(b, k)),A^2 -> H(U1(c, a), U1(b, k))) ---------------------------------------- (248) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(b, d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (249) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(b, d)),A^2 -> H(U1(e, a), U1(b, d))) (A^2 -> H(U1(k, a), U1(b, d)),A^2 -> H(U1(k, a), U1(b, d))) (A^2 -> H(U1(c, c), U1(b, d)),A^2 -> H(U1(c, c), U1(b, d))) (A^2 -> H(U1(c, d), U1(b, d)),A^2 -> H(U1(c, d), U1(b, d))) (A^2 -> H(U1(c, a), U1(c, d)),A^2 -> H(U1(c, a), U1(c, d))) (A^2 -> H(U1(c, a), U1(d, d)),A^2 -> H(U1(c, a), U1(d, d))) (A^2 -> H(U1(c, a), U1(b, k)),A^2 -> H(U1(c, a), U1(b, k))) ---------------------------------------- (250) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(c, c)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (251) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(c, c)),A^2 -> H(U1(e, a), U1(c, c))) (A^2 -> H(U1(k, a), U1(c, c)),A^2 -> H(U1(k, a), U1(c, c))) (A^2 -> H(U1(c, c), U1(c, c)),A^2 -> H(U1(c, c), U1(c, c))) (A^2 -> H(U1(c, d), U1(c, c)),A^2 -> H(U1(c, d), U1(c, c))) (A^2 -> H(U1(c, a), U1(e, c)),A^2 -> H(U1(c, a), U1(e, c))) (A^2 -> H(U1(c, a), U1(k, c)),A^2 -> H(U1(c, a), U1(k, c))) (A^2 -> H(U1(c, a), U1(c, e)),A^2 -> H(U1(c, a), U1(c, e))) (A^2 -> H(U1(c, a), U1(c, k)),A^2 -> H(U1(c, a), U1(c, k))) ---------------------------------------- (252) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), f(e)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (253) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), f(e)),A^2 -> H(U1(e, a), f(e))) (A^2 -> H(U1(k, a), f(e)),A^2 -> H(U1(k, a), f(e))) (A^2 -> H(U1(c, c), f(e)),A^2 -> H(U1(c, c), f(e))) (A^2 -> H(U1(c, d), f(e)),A^2 -> H(U1(c, d), f(e))) (A^2 -> H(U1(c, a), U1(e, e)),A^2 -> H(U1(c, a), U1(e, e))) ---------------------------------------- (254) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), f(k)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (255) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), f(k)),A^2 -> H(U1(e, a), f(k))) (A^2 -> H(U1(k, a), f(k)),A^2 -> H(U1(k, a), f(k))) (A^2 -> H(U1(c, c), f(k)),A^2 -> H(U1(c, c), f(k))) (A^2 -> H(U1(c, d), f(k)),A^2 -> H(U1(c, d), f(k))) (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) ---------------------------------------- (256) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, a), U1(d, d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (257) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(d, d)),A^2 -> H(U1(e, a), U1(d, d))) (A^2 -> H(U1(k, a), U1(d, d)),A^2 -> H(U1(k, a), U1(d, d))) (A^2 -> H(U1(c, c), U1(d, d)),A^2 -> H(U1(c, c), U1(d, d))) (A^2 -> H(U1(c, d), U1(d, d)),A^2 -> H(U1(c, d), U1(d, d))) (A^2 -> H(U1(c, a), U1(k, d)),A^2 -> H(U1(c, a), U1(k, d))) (A^2 -> H(U1(c, a), U1(d, k)),A^2 -> H(U1(c, a), U1(d, k))) ---------------------------------------- (258) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, e), f(b)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (259) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(b)),A^2 -> H(U1(k, e), f(b))) (A^2 -> H(U1(d, e), U1(b, b)),A^2 -> H(U1(d, e), U1(b, b))) (A^2 -> H(U1(d, e), f(c)),A^2 -> H(U1(d, e), f(c))) (A^2 -> H(U1(d, e), f(d)),A^2 -> H(U1(d, e), f(d))) ---------------------------------------- (260) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, c), U1(b, b)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (261) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, b)),A^2 -> H(U1(k, c), U1(b, b))) (A^2 -> H(U1(d, e), U1(b, b)),A^2 -> H(U1(d, e), U1(b, b))) (A^2 -> H(U1(d, k), U1(b, b)),A^2 -> H(U1(d, k), U1(b, b))) (A^2 -> H(U1(d, c), U1(c, b)),A^2 -> H(U1(d, c), U1(c, b))) (A^2 -> H(U1(d, c), U1(d, b)),A^2 -> H(U1(d, c), U1(d, b))) (A^2 -> H(U1(d, c), U1(b, c)),A^2 -> H(U1(d, c), U1(b, c))) (A^2 -> H(U1(d, c), U1(b, d)),A^2 -> H(U1(d, c), U1(b, d))) ---------------------------------------- (262) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, c), f(c)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (263) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(c)),A^2 -> H(U1(k, c), f(c))) (A^2 -> H(U1(d, e), f(c)),A^2 -> H(U1(d, e), f(c))) (A^2 -> H(U1(d, k), f(c)),A^2 -> H(U1(d, k), f(c))) (A^2 -> H(U1(d, c), U1(c, c)),A^2 -> H(U1(d, c), U1(c, c))) (A^2 -> H(U1(d, c), f(e)),A^2 -> H(U1(d, c), f(e))) (A^2 -> H(U1(d, c), f(k)),A^2 -> H(U1(d, c), f(k))) ---------------------------------------- (264) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, c), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (265) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(d)),A^2 -> H(U1(k, c), f(d))) (A^2 -> H(U1(d, e), f(d)),A^2 -> H(U1(d, e), f(d))) (A^2 -> H(U1(d, k), f(d)),A^2 -> H(U1(d, k), f(d))) (A^2 -> H(U1(d, c), U1(d, d)),A^2 -> H(U1(d, c), U1(d, d))) (A^2 -> H(U1(d, c), f(k)),A^2 -> H(U1(d, c), f(k))) ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(c, b)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (267) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(c, b)),A^2 -> H(U1(k, a), U1(c, b))) (A^2 -> H(U1(d, c), U1(c, b)),A^2 -> H(U1(d, c), U1(c, b))) (A^2 -> H(U1(d, d), U1(c, b)),A^2 -> H(U1(d, d), U1(c, b))) (A^2 -> H(U1(d, a), U1(e, b)),A^2 -> H(U1(d, a), U1(e, b))) (A^2 -> H(U1(d, a), U1(k, b)),A^2 -> H(U1(d, a), U1(k, b))) (A^2 -> H(U1(d, a), U1(c, c)),A^2 -> H(U1(d, a), U1(c, c))) (A^2 -> H(U1(d, a), U1(c, d)),A^2 -> H(U1(d, a), U1(c, d))) ---------------------------------------- (268) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(d, b)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (269) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(d, b)),A^2 -> H(U1(k, a), U1(d, b))) (A^2 -> H(U1(d, c), U1(d, b)),A^2 -> H(U1(d, c), U1(d, b))) (A^2 -> H(U1(d, d), U1(d, b)),A^2 -> H(U1(d, d), U1(d, b))) (A^2 -> H(U1(d, a), U1(k, b)),A^2 -> H(U1(d, a), U1(k, b))) (A^2 -> H(U1(d, a), U1(d, c)),A^2 -> H(U1(d, a), U1(d, c))) (A^2 -> H(U1(d, a), U1(d, d)),A^2 -> H(U1(d, a), U1(d, d))) ---------------------------------------- (270) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(b, c)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (271) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(b, c)),A^2 -> H(U1(k, a), U1(b, c))) (A^2 -> H(U1(d, c), U1(b, c)),A^2 -> H(U1(d, c), U1(b, c))) (A^2 -> H(U1(d, d), U1(b, c)),A^2 -> H(U1(d, d), U1(b, c))) (A^2 -> H(U1(d, a), U1(c, c)),A^2 -> H(U1(d, a), U1(c, c))) (A^2 -> H(U1(d, a), U1(d, c)),A^2 -> H(U1(d, a), U1(d, c))) (A^2 -> H(U1(d, a), U1(b, e)),A^2 -> H(U1(d, a), U1(b, e))) (A^2 -> H(U1(d, a), U1(b, k)),A^2 -> H(U1(d, a), U1(b, k))) ---------------------------------------- (272) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(b, d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (273) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(b, d)),A^2 -> H(U1(k, a), U1(b, d))) (A^2 -> H(U1(d, c), U1(b, d)),A^2 -> H(U1(d, c), U1(b, d))) (A^2 -> H(U1(d, d), U1(b, d)),A^2 -> H(U1(d, d), U1(b, d))) (A^2 -> H(U1(d, a), U1(c, d)),A^2 -> H(U1(d, a), U1(c, d))) (A^2 -> H(U1(d, a), U1(d, d)),A^2 -> H(U1(d, a), U1(d, d))) (A^2 -> H(U1(d, a), U1(b, k)),A^2 -> H(U1(d, a), U1(b, k))) ---------------------------------------- (274) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(c, c)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (275) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(c, c)),A^2 -> H(U1(k, a), U1(c, c))) (A^2 -> H(U1(d, c), U1(c, c)),A^2 -> H(U1(d, c), U1(c, c))) (A^2 -> H(U1(d, d), U1(c, c)),A^2 -> H(U1(d, d), U1(c, c))) (A^2 -> H(U1(d, a), U1(e, c)),A^2 -> H(U1(d, a), U1(e, c))) (A^2 -> H(U1(d, a), U1(k, c)),A^2 -> H(U1(d, a), U1(k, c))) (A^2 -> H(U1(d, a), U1(c, e)),A^2 -> H(U1(d, a), U1(c, e))) (A^2 -> H(U1(d, a), U1(c, k)),A^2 -> H(U1(d, a), U1(c, k))) ---------------------------------------- (276) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), f(e)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (277) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), f(e)),A^2 -> H(U1(k, a), f(e))) (A^2 -> H(U1(d, c), f(e)),A^2 -> H(U1(d, c), f(e))) (A^2 -> H(U1(d, d), f(e)),A^2 -> H(U1(d, d), f(e))) (A^2 -> H(U1(d, a), U1(e, e)),A^2 -> H(U1(d, a), U1(e, e))) ---------------------------------------- (278) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), f(k)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (279) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), f(k)),A^2 -> H(U1(k, a), f(k))) (A^2 -> H(U1(d, c), f(k)),A^2 -> H(U1(d, c), f(k))) (A^2 -> H(U1(d, d), f(k)),A^2 -> H(U1(d, d), f(k))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) ---------------------------------------- (280) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, a), U1(d, d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (281) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(d, d)),A^2 -> H(U1(k, a), U1(d, d))) (A^2 -> H(U1(d, c), U1(d, d)),A^2 -> H(U1(d, c), U1(d, d))) (A^2 -> H(U1(d, d), U1(d, d)),A^2 -> H(U1(d, d), U1(d, d))) (A^2 -> H(U1(d, a), U1(k, d)),A^2 -> H(U1(d, a), U1(k, d))) (A^2 -> H(U1(d, a), U1(d, k)),A^2 -> H(U1(d, a), U1(d, k))) ---------------------------------------- (282) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, e), U1(b, b)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (283) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(b, b)),A^2 -> H(U1(c, e), U1(b, b))) (A^2 -> H(U1(d, e), U1(b, b)),A^2 -> H(U1(d, e), U1(b, b))) (A^2 -> H(U1(a, e), U1(c, b)),A^2 -> H(U1(a, e), U1(c, b))) (A^2 -> H(U1(a, e), U1(d, b)),A^2 -> H(U1(a, e), U1(d, b))) (A^2 -> H(U1(a, e), U1(b, c)),A^2 -> H(U1(a, e), U1(b, c))) (A^2 -> H(U1(a, e), U1(b, d)),A^2 -> H(U1(a, e), U1(b, d))) ---------------------------------------- (284) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, e), f(c)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (285) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), f(c)),A^2 -> H(U1(c, e), f(c))) (A^2 -> H(U1(d, e), f(c)),A^2 -> H(U1(d, e), f(c))) (A^2 -> H(U1(a, e), U1(c, c)),A^2 -> H(U1(a, e), U1(c, c))) (A^2 -> H(U1(a, e), f(e)),A^2 -> H(U1(a, e), f(e))) (A^2 -> H(U1(a, e), f(k)),A^2 -> H(U1(a, e), f(k))) ---------------------------------------- (286) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, e), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (287) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), f(d)),A^2 -> H(U1(c, e), f(d))) (A^2 -> H(U1(d, e), f(d)),A^2 -> H(U1(d, e), f(d))) (A^2 -> H(U1(a, e), U1(d, d)),A^2 -> H(U1(a, e), U1(d, d))) (A^2 -> H(U1(a, e), f(k)),A^2 -> H(U1(a, e), f(k))) ---------------------------------------- (288) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, k), U1(b, b)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (289) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(b, b)),A^2 -> H(U1(c, k), U1(b, b))) (A^2 -> H(U1(d, k), U1(b, b)),A^2 -> H(U1(d, k), U1(b, b))) (A^2 -> H(U1(a, k), U1(c, b)),A^2 -> H(U1(a, k), U1(c, b))) (A^2 -> H(U1(a, k), U1(d, b)),A^2 -> H(U1(a, k), U1(d, b))) (A^2 -> H(U1(a, k), U1(b, c)),A^2 -> H(U1(a, k), U1(b, c))) (A^2 -> H(U1(a, k), U1(b, d)),A^2 -> H(U1(a, k), U1(b, d))) ---------------------------------------- (290) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, k), f(c)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (291) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), f(c)),A^2 -> H(U1(c, k), f(c))) (A^2 -> H(U1(d, k), f(c)),A^2 -> H(U1(d, k), f(c))) (A^2 -> H(U1(a, k), U1(c, c)),A^2 -> H(U1(a, k), U1(c, c))) (A^2 -> H(U1(a, k), f(e)),A^2 -> H(U1(a, k), f(e))) (A^2 -> H(U1(a, k), f(k)),A^2 -> H(U1(a, k), f(k))) ---------------------------------------- (292) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, k), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (293) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), f(d)),A^2 -> H(U1(c, k), f(d))) (A^2 -> H(U1(d, k), f(d)),A^2 -> H(U1(d, k), f(d))) (A^2 -> H(U1(a, k), U1(d, d)),A^2 -> H(U1(a, k), U1(d, d))) (A^2 -> H(U1(a, k), f(k)),A^2 -> H(U1(a, k), f(k))) ---------------------------------------- (294) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(c, b)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (295) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, b)),A^2 -> H(U1(c, c), U1(c, b))) (A^2 -> H(U1(d, c), U1(c, b)),A^2 -> H(U1(d, c), U1(c, b))) (A^2 -> H(U1(a, e), U1(c, b)),A^2 -> H(U1(a, e), U1(c, b))) (A^2 -> H(U1(a, k), U1(c, b)),A^2 -> H(U1(a, k), U1(c, b))) (A^2 -> H(U1(a, c), U1(e, b)),A^2 -> H(U1(a, c), U1(e, b))) (A^2 -> H(U1(a, c), U1(k, b)),A^2 -> H(U1(a, c), U1(k, b))) (A^2 -> H(U1(a, c), U1(c, c)),A^2 -> H(U1(a, c), U1(c, c))) (A^2 -> H(U1(a, c), U1(c, d)),A^2 -> H(U1(a, c), U1(c, d))) ---------------------------------------- (296) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(d, b)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (297) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, b)),A^2 -> H(U1(c, c), U1(d, b))) (A^2 -> H(U1(d, c), U1(d, b)),A^2 -> H(U1(d, c), U1(d, b))) (A^2 -> H(U1(a, e), U1(d, b)),A^2 -> H(U1(a, e), U1(d, b))) (A^2 -> H(U1(a, k), U1(d, b)),A^2 -> H(U1(a, k), U1(d, b))) (A^2 -> H(U1(a, c), U1(k, b)),A^2 -> H(U1(a, c), U1(k, b))) (A^2 -> H(U1(a, c), U1(d, c)),A^2 -> H(U1(a, c), U1(d, c))) (A^2 -> H(U1(a, c), U1(d, d)),A^2 -> H(U1(a, c), U1(d, d))) ---------------------------------------- (298) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(b, c)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (299) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, c)),A^2 -> H(U1(c, c), U1(b, c))) (A^2 -> H(U1(d, c), U1(b, c)),A^2 -> H(U1(d, c), U1(b, c))) (A^2 -> H(U1(a, e), U1(b, c)),A^2 -> H(U1(a, e), U1(b, c))) (A^2 -> H(U1(a, k), U1(b, c)),A^2 -> H(U1(a, k), U1(b, c))) (A^2 -> H(U1(a, c), U1(c, c)),A^2 -> H(U1(a, c), U1(c, c))) (A^2 -> H(U1(a, c), U1(d, c)),A^2 -> H(U1(a, c), U1(d, c))) (A^2 -> H(U1(a, c), U1(b, e)),A^2 -> H(U1(a, c), U1(b, e))) (A^2 -> H(U1(a, c), U1(b, k)),A^2 -> H(U1(a, c), U1(b, k))) ---------------------------------------- (300) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(b, d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (301) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, d)),A^2 -> H(U1(c, c), U1(b, d))) (A^2 -> H(U1(d, c), U1(b, d)),A^2 -> H(U1(d, c), U1(b, d))) (A^2 -> H(U1(a, e), U1(b, d)),A^2 -> H(U1(a, e), U1(b, d))) (A^2 -> H(U1(a, k), U1(b, d)),A^2 -> H(U1(a, k), U1(b, d))) (A^2 -> H(U1(a, c), U1(c, d)),A^2 -> H(U1(a, c), U1(c, d))) (A^2 -> H(U1(a, c), U1(d, d)),A^2 -> H(U1(a, c), U1(d, d))) (A^2 -> H(U1(a, c), U1(b, k)),A^2 -> H(U1(a, c), U1(b, k))) ---------------------------------------- (302) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(c, c)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (303) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, c)),A^2 -> H(U1(c, c), U1(c, c))) (A^2 -> H(U1(d, c), U1(c, c)),A^2 -> H(U1(d, c), U1(c, c))) (A^2 -> H(U1(a, e), U1(c, c)),A^2 -> H(U1(a, e), U1(c, c))) (A^2 -> H(U1(a, k), U1(c, c)),A^2 -> H(U1(a, k), U1(c, c))) (A^2 -> H(U1(a, c), U1(e, c)),A^2 -> H(U1(a, c), U1(e, c))) (A^2 -> H(U1(a, c), U1(k, c)),A^2 -> H(U1(a, c), U1(k, c))) (A^2 -> H(U1(a, c), U1(c, e)),A^2 -> H(U1(a, c), U1(c, e))) (A^2 -> H(U1(a, c), U1(c, k)),A^2 -> H(U1(a, c), U1(c, k))) ---------------------------------------- (304) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), f(e)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (305) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(e)),A^2 -> H(U1(c, c), f(e))) (A^2 -> H(U1(d, c), f(e)),A^2 -> H(U1(d, c), f(e))) (A^2 -> H(U1(a, e), f(e)),A^2 -> H(U1(a, e), f(e))) (A^2 -> H(U1(a, k), f(e)),A^2 -> H(U1(a, k), f(e))) (A^2 -> H(U1(a, c), U1(e, e)),A^2 -> H(U1(a, c), U1(e, e))) ---------------------------------------- (306) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), f(k)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (307) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), f(k)),A^2 -> H(U1(c, c), f(k))) (A^2 -> H(U1(d, c), f(k)),A^2 -> H(U1(d, c), f(k))) (A^2 -> H(U1(a, e), f(k)),A^2 -> H(U1(a, e), f(k))) (A^2 -> H(U1(a, k), f(k)),A^2 -> H(U1(a, k), f(k))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) ---------------------------------------- (308) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, c), U1(d, d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (309) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, d)),A^2 -> H(U1(c, c), U1(d, d))) (A^2 -> H(U1(d, c), U1(d, d)),A^2 -> H(U1(d, c), U1(d, d))) (A^2 -> H(U1(a, e), U1(d, d)),A^2 -> H(U1(a, e), U1(d, d))) (A^2 -> H(U1(a, k), U1(d, d)),A^2 -> H(U1(a, k), U1(d, d))) (A^2 -> H(U1(a, c), U1(k, d)),A^2 -> H(U1(a, c), U1(k, d))) (A^2 -> H(U1(a, c), U1(d, k)),A^2 -> H(U1(a, c), U1(d, k))) ---------------------------------------- (310) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(c, b)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (311) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(c, b)),A^2 -> H(U1(c, d), U1(c, b))) (A^2 -> H(U1(d, d), U1(c, b)),A^2 -> H(U1(d, d), U1(c, b))) (A^2 -> H(U1(a, k), U1(c, b)),A^2 -> H(U1(a, k), U1(c, b))) (A^2 -> H(U1(a, d), U1(e, b)),A^2 -> H(U1(a, d), U1(e, b))) (A^2 -> H(U1(a, d), U1(k, b)),A^2 -> H(U1(a, d), U1(k, b))) (A^2 -> H(U1(a, d), U1(c, c)),A^2 -> H(U1(a, d), U1(c, c))) (A^2 -> H(U1(a, d), U1(c, d)),A^2 -> H(U1(a, d), U1(c, d))) ---------------------------------------- (312) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(d, b)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (313) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(d, b)),A^2 -> H(U1(c, d), U1(d, b))) (A^2 -> H(U1(d, d), U1(d, b)),A^2 -> H(U1(d, d), U1(d, b))) (A^2 -> H(U1(a, k), U1(d, b)),A^2 -> H(U1(a, k), U1(d, b))) (A^2 -> H(U1(a, d), U1(k, b)),A^2 -> H(U1(a, d), U1(k, b))) (A^2 -> H(U1(a, d), U1(d, c)),A^2 -> H(U1(a, d), U1(d, c))) (A^2 -> H(U1(a, d), U1(d, d)),A^2 -> H(U1(a, d), U1(d, d))) ---------------------------------------- (314) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(b, c)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (315) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(b, c)),A^2 -> H(U1(c, d), U1(b, c))) (A^2 -> H(U1(d, d), U1(b, c)),A^2 -> H(U1(d, d), U1(b, c))) (A^2 -> H(U1(a, k), U1(b, c)),A^2 -> H(U1(a, k), U1(b, c))) (A^2 -> H(U1(a, d), U1(c, c)),A^2 -> H(U1(a, d), U1(c, c))) (A^2 -> H(U1(a, d), U1(d, c)),A^2 -> H(U1(a, d), U1(d, c))) (A^2 -> H(U1(a, d), U1(b, e)),A^2 -> H(U1(a, d), U1(b, e))) (A^2 -> H(U1(a, d), U1(b, k)),A^2 -> H(U1(a, d), U1(b, k))) ---------------------------------------- (316) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(b, d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (317) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(b, d)),A^2 -> H(U1(c, d), U1(b, d))) (A^2 -> H(U1(d, d), U1(b, d)),A^2 -> H(U1(d, d), U1(b, d))) (A^2 -> H(U1(a, k), U1(b, d)),A^2 -> H(U1(a, k), U1(b, d))) (A^2 -> H(U1(a, d), U1(c, d)),A^2 -> H(U1(a, d), U1(c, d))) (A^2 -> H(U1(a, d), U1(d, d)),A^2 -> H(U1(a, d), U1(d, d))) (A^2 -> H(U1(a, d), U1(b, k)),A^2 -> H(U1(a, d), U1(b, k))) ---------------------------------------- (318) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(c, c)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (319) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(c, c)),A^2 -> H(U1(c, d), U1(c, c))) (A^2 -> H(U1(d, d), U1(c, c)),A^2 -> H(U1(d, d), U1(c, c))) (A^2 -> H(U1(a, k), U1(c, c)),A^2 -> H(U1(a, k), U1(c, c))) (A^2 -> H(U1(a, d), U1(e, c)),A^2 -> H(U1(a, d), U1(e, c))) (A^2 -> H(U1(a, d), U1(k, c)),A^2 -> H(U1(a, d), U1(k, c))) (A^2 -> H(U1(a, d), U1(c, e)),A^2 -> H(U1(a, d), U1(c, e))) (A^2 -> H(U1(a, d), U1(c, k)),A^2 -> H(U1(a, d), U1(c, k))) ---------------------------------------- (320) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), f(e)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (321) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), f(e)),A^2 -> H(U1(c, d), f(e))) (A^2 -> H(U1(d, d), f(e)),A^2 -> H(U1(d, d), f(e))) (A^2 -> H(U1(a, k), f(e)),A^2 -> H(U1(a, k), f(e))) (A^2 -> H(U1(a, d), U1(e, e)),A^2 -> H(U1(a, d), U1(e, e))) ---------------------------------------- (322) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), f(k)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (323) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), f(k)),A^2 -> H(U1(c, d), f(k))) (A^2 -> H(U1(d, d), f(k)),A^2 -> H(U1(d, d), f(k))) (A^2 -> H(U1(a, k), f(k)),A^2 -> H(U1(a, k), f(k))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (324) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (325) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(d, d)),A^2 -> H(U1(c, d), U1(d, d))) (A^2 -> H(U1(d, d), U1(d, d)),A^2 -> H(U1(d, d), U1(d, d))) (A^2 -> H(U1(a, k), U1(d, d)),A^2 -> H(U1(a, k), U1(d, d))) (A^2 -> H(U1(a, d), U1(k, d)),A^2 -> H(U1(a, d), U1(k, d))) (A^2 -> H(U1(a, d), U1(d, k)),A^2 -> H(U1(a, d), U1(d, k))) ---------------------------------------- (326) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(e, b)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (327) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(e, b)),A^2 -> H(U1(c, a), U1(e, b))) (A^2 -> H(U1(d, a), U1(e, b)),A^2 -> H(U1(d, a), U1(e, b))) (A^2 -> H(U1(a, c), U1(e, b)),A^2 -> H(U1(a, c), U1(e, b))) (A^2 -> H(U1(a, d), U1(e, b)),A^2 -> H(U1(a, d), U1(e, b))) (A^2 -> H(U1(a, a), b),A^2 -> H(U1(a, a), b)) (A^2 -> H(U1(a, a), U1(e, c)),A^2 -> H(U1(a, a), U1(e, c))) (A^2 -> H(U1(a, a), U1(e, d)),A^2 -> H(U1(a, a), U1(e, d))) ---------------------------------------- (328) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(k, b)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (329) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(k, b)),A^2 -> H(U1(c, a), U1(k, b))) (A^2 -> H(U1(d, a), U1(k, b)),A^2 -> H(U1(d, a), U1(k, b))) (A^2 -> H(U1(a, c), U1(k, b)),A^2 -> H(U1(a, c), U1(k, b))) (A^2 -> H(U1(a, d), U1(k, b)),A^2 -> H(U1(a, d), U1(k, b))) (A^2 -> H(U1(a, a), U1(k, c)),A^2 -> H(U1(a, a), U1(k, c))) (A^2 -> H(U1(a, a), U1(k, d)),A^2 -> H(U1(a, a), U1(k, d))) ---------------------------------------- (330) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(c, d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (331) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(c, d)),A^2 -> H(U1(c, a), U1(c, d))) (A^2 -> H(U1(d, a), U1(c, d)),A^2 -> H(U1(d, a), U1(c, d))) (A^2 -> H(U1(a, c), U1(c, d)),A^2 -> H(U1(a, c), U1(c, d))) (A^2 -> H(U1(a, d), U1(c, d)),A^2 -> H(U1(a, d), U1(c, d))) (A^2 -> H(U1(a, a), U1(e, d)),A^2 -> H(U1(a, a), U1(e, d))) (A^2 -> H(U1(a, a), U1(k, d)),A^2 -> H(U1(a, a), U1(k, d))) (A^2 -> H(U1(a, a), U1(c, k)),A^2 -> H(U1(a, a), U1(c, k))) ---------------------------------------- (332) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(d, c)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (333) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(d, c)),A^2 -> H(U1(c, a), U1(d, c))) (A^2 -> H(U1(d, a), U1(d, c)),A^2 -> H(U1(d, a), U1(d, c))) (A^2 -> H(U1(a, c), U1(d, c)),A^2 -> H(U1(a, c), U1(d, c))) (A^2 -> H(U1(a, d), U1(d, c)),A^2 -> H(U1(a, d), U1(d, c))) (A^2 -> H(U1(a, a), U1(k, c)),A^2 -> H(U1(a, a), U1(k, c))) (A^2 -> H(U1(a, a), U1(d, e)),A^2 -> H(U1(a, a), U1(d, e))) (A^2 -> H(U1(a, a), U1(d, k)),A^2 -> H(U1(a, a), U1(d, k))) ---------------------------------------- (334) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(b, e)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (335) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(b, e)),A^2 -> H(U1(c, a), U1(b, e))) (A^2 -> H(U1(d, a), U1(b, e)),A^2 -> H(U1(d, a), U1(b, e))) (A^2 -> H(U1(a, c), U1(b, e)),A^2 -> H(U1(a, c), U1(b, e))) (A^2 -> H(U1(a, d), U1(b, e)),A^2 -> H(U1(a, d), U1(b, e))) (A^2 -> H(U1(a, a), U1(c, e)),A^2 -> H(U1(a, a), U1(c, e))) (A^2 -> H(U1(a, a), U1(d, e)),A^2 -> H(U1(a, a), U1(d, e))) ---------------------------------------- (336) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(b, k)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (337) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(b, k)),A^2 -> H(U1(c, a), U1(b, k))) (A^2 -> H(U1(d, a), U1(b, k)),A^2 -> H(U1(d, a), U1(b, k))) (A^2 -> H(U1(a, c), U1(b, k)),A^2 -> H(U1(a, c), U1(b, k))) (A^2 -> H(U1(a, d), U1(b, k)),A^2 -> H(U1(a, d), U1(b, k))) (A^2 -> H(U1(a, a), U1(c, k)),A^2 -> H(U1(a, a), U1(c, k))) (A^2 -> H(U1(a, a), U1(d, k)),A^2 -> H(U1(a, a), U1(d, k))) ---------------------------------------- (338) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(e, c)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (339) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(e, c)),A^2 -> H(U1(c, a), U1(e, c))) (A^2 -> H(U1(d, a), U1(e, c)),A^2 -> H(U1(d, a), U1(e, c))) (A^2 -> H(U1(a, c), U1(e, c)),A^2 -> H(U1(a, c), U1(e, c))) (A^2 -> H(U1(a, d), U1(e, c)),A^2 -> H(U1(a, d), U1(e, c))) (A^2 -> H(U1(a, a), c),A^2 -> H(U1(a, a), c)) (A^2 -> H(U1(a, a), U1(e, e)),A^2 -> H(U1(a, a), U1(e, e))) (A^2 -> H(U1(a, a), U1(e, k)),A^2 -> H(U1(a, a), U1(e, k))) ---------------------------------------- (340) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(k, c)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (341) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(k, c)),A^2 -> H(U1(c, a), U1(k, c))) (A^2 -> H(U1(d, a), U1(k, c)),A^2 -> H(U1(d, a), U1(k, c))) (A^2 -> H(U1(a, c), U1(k, c)),A^2 -> H(U1(a, c), U1(k, c))) (A^2 -> H(U1(a, d), U1(k, c)),A^2 -> H(U1(a, d), U1(k, c))) (A^2 -> H(U1(a, a), U1(k, e)),A^2 -> H(U1(a, a), U1(k, e))) (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) ---------------------------------------- (342) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(c, e)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (343) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(c, e)),A^2 -> H(U1(c, a), U1(c, e))) (A^2 -> H(U1(d, a), U1(c, e)),A^2 -> H(U1(d, a), U1(c, e))) (A^2 -> H(U1(a, c), U1(c, e)),A^2 -> H(U1(a, c), U1(c, e))) (A^2 -> H(U1(a, d), U1(c, e)),A^2 -> H(U1(a, d), U1(c, e))) (A^2 -> H(U1(a, a), U1(e, e)),A^2 -> H(U1(a, a), U1(e, e))) (A^2 -> H(U1(a, a), U1(k, e)),A^2 -> H(U1(a, a), U1(k, e))) ---------------------------------------- (344) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(c, k)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (345) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(c, k)),A^2 -> H(U1(c, a), U1(c, k))) (A^2 -> H(U1(d, a), U1(c, k)),A^2 -> H(U1(d, a), U1(c, k))) (A^2 -> H(U1(a, c), U1(c, k)),A^2 -> H(U1(a, c), U1(c, k))) (A^2 -> H(U1(a, d), U1(c, k)),A^2 -> H(U1(a, d), U1(c, k))) (A^2 -> H(U1(a, a), U1(e, k)),A^2 -> H(U1(a, a), U1(e, k))) (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) ---------------------------------------- (346) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(e, e)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (347) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(e, e)),A^2 -> H(U1(c, a), U1(e, e))) (A^2 -> H(U1(d, a), U1(e, e)),A^2 -> H(U1(d, a), U1(e, e))) (A^2 -> H(U1(a, c), U1(e, e)),A^2 -> H(U1(a, c), U1(e, e))) (A^2 -> H(U1(a, d), U1(e, e)),A^2 -> H(U1(a, d), U1(e, e))) (A^2 -> H(U1(a, a), e),A^2 -> H(U1(a, a), e)) ---------------------------------------- (348) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (349) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (350) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(k, d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (351) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(k, d)),A^2 -> H(U1(c, a), U1(k, d))) (A^2 -> H(U1(d, a), U1(k, d)),A^2 -> H(U1(d, a), U1(k, d))) (A^2 -> H(U1(a, c), U1(k, d)),A^2 -> H(U1(a, c), U1(k, d))) (A^2 -> H(U1(a, d), U1(k, d)),A^2 -> H(U1(a, d), U1(k, d))) (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) ---------------------------------------- (352) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(a, a), U1(d, k)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (353) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(d, k)),A^2 -> H(U1(c, a), U1(d, k))) (A^2 -> H(U1(d, a), U1(d, k)),A^2 -> H(U1(d, a), U1(d, k))) (A^2 -> H(U1(a, c), U1(d, k)),A^2 -> H(U1(a, c), U1(d, k))) (A^2 -> H(U1(a, d), U1(d, k)),A^2 -> H(U1(a, d), U1(d, k))) (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) ---------------------------------------- (354) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(c, f(b)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (355) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(b)),A^2 -> H(e, f(b))) (A^2 -> H(k, f(b)),A^2 -> H(k, f(b))) (A^2 -> H(c, U1(b, b)),A^2 -> H(c, U1(b, b))) (A^2 -> H(c, f(c)),A^2 -> H(c, f(c))) (A^2 -> H(c, f(d)),A^2 -> H(c, f(d))) ---------------------------------------- (356) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, k), f(b)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (357) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(b)),A^2 -> H(k, f(b))) (A^2 -> H(U1(e, k), U1(b, b)),A^2 -> H(U1(e, k), U1(b, b))) (A^2 -> H(U1(e, k), f(c)),A^2 -> H(U1(e, k), f(c))) (A^2 -> H(U1(e, k), f(d)),A^2 -> H(U1(e, k), f(d))) ---------------------------------------- (358) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, c), U1(b, b)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (359) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, b)),A^2 -> H(c, U1(b, b))) (A^2 -> H(U1(e, e), U1(b, b)),A^2 -> H(U1(e, e), U1(b, b))) (A^2 -> H(U1(e, k), U1(b, b)),A^2 -> H(U1(e, k), U1(b, b))) (A^2 -> H(U1(e, c), U1(c, b)),A^2 -> H(U1(e, c), U1(c, b))) (A^2 -> H(U1(e, c), U1(d, b)),A^2 -> H(U1(e, c), U1(d, b))) (A^2 -> H(U1(e, c), U1(b, c)),A^2 -> H(U1(e, c), U1(b, c))) (A^2 -> H(U1(e, c), U1(b, d)),A^2 -> H(U1(e, c), U1(b, d))) ---------------------------------------- (360) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, c), f(c)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (361) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(c)),A^2 -> H(c, f(c))) (A^2 -> H(U1(e, e), f(c)),A^2 -> H(U1(e, e), f(c))) (A^2 -> H(U1(e, k), f(c)),A^2 -> H(U1(e, k), f(c))) (A^2 -> H(U1(e, c), U1(c, c)),A^2 -> H(U1(e, c), U1(c, c))) (A^2 -> H(U1(e, c), f(e)),A^2 -> H(U1(e, c), f(e))) (A^2 -> H(U1(e, c), f(k)),A^2 -> H(U1(e, c), f(k))) ---------------------------------------- (362) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(e, c), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (363) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(d)),A^2 -> H(c, f(d))) (A^2 -> H(U1(e, e), f(d)),A^2 -> H(U1(e, e), f(d))) (A^2 -> H(U1(e, k), f(d)),A^2 -> H(U1(e, k), f(d))) (A^2 -> H(U1(e, c), U1(d, d)),A^2 -> H(U1(e, c), U1(d, d))) (A^2 -> H(U1(e, c), f(k)),A^2 -> H(U1(e, c), f(k))) ---------------------------------------- (364) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, e), f(b)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (365) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, b)),A^2 -> H(U1(k, e), U1(b, b))) (A^2 -> H(U1(k, e), f(c)),A^2 -> H(U1(k, e), f(c))) (A^2 -> H(U1(k, e), f(d)),A^2 -> H(U1(k, e), f(d))) ---------------------------------------- (366) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, c), U1(b, b)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (367) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, b)),A^2 -> H(U1(k, e), U1(b, b))) (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(U1(k, c), U1(c, b)),A^2 -> H(U1(k, c), U1(c, b))) (A^2 -> H(U1(k, c), U1(d, b)),A^2 -> H(U1(k, c), U1(d, b))) (A^2 -> H(U1(k, c), U1(b, c)),A^2 -> H(U1(k, c), U1(b, c))) (A^2 -> H(U1(k, c), U1(b, d)),A^2 -> H(U1(k, c), U1(b, d))) ---------------------------------------- (368) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, c), f(c)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (369) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(c)),A^2 -> H(U1(k, e), f(c))) (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(U1(k, c), U1(c, c)),A^2 -> H(U1(k, c), U1(c, c))) (A^2 -> H(U1(k, c), f(e)),A^2 -> H(U1(k, c), f(e))) (A^2 -> H(U1(k, c), f(k)),A^2 -> H(U1(k, c), f(k))) ---------------------------------------- (370) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(k, c), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (371) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(d)),A^2 -> H(U1(k, e), f(d))) (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) (A^2 -> H(U1(k, c), U1(d, d)),A^2 -> H(U1(k, c), U1(d, d))) (A^2 -> H(U1(k, c), f(k)),A^2 -> H(U1(k, c), f(k))) ---------------------------------------- (372) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, e), U1(b, b)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (373) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, b)),A^2 -> H(U1(e, e), U1(b, b))) (A^2 -> H(U1(k, e), U1(b, b)),A^2 -> H(U1(k, e), U1(b, b))) (A^2 -> H(U1(c, e), U1(c, b)),A^2 -> H(U1(c, e), U1(c, b))) (A^2 -> H(U1(c, e), U1(d, b)),A^2 -> H(U1(c, e), U1(d, b))) (A^2 -> H(U1(c, e), U1(b, c)),A^2 -> H(U1(c, e), U1(b, c))) (A^2 -> H(U1(c, e), U1(b, d)),A^2 -> H(U1(c, e), U1(b, d))) ---------------------------------------- (374) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, e), f(c)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (375) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(c)),A^2 -> H(U1(e, e), f(c))) (A^2 -> H(U1(k, e), f(c)),A^2 -> H(U1(k, e), f(c))) (A^2 -> H(U1(c, e), U1(c, c)),A^2 -> H(U1(c, e), U1(c, c))) (A^2 -> H(U1(c, e), f(e)),A^2 -> H(U1(c, e), f(e))) (A^2 -> H(U1(c, e), f(k)),A^2 -> H(U1(c, e), f(k))) ---------------------------------------- (376) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, e), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (377) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(d)),A^2 -> H(U1(e, e), f(d))) (A^2 -> H(U1(k, e), f(d)),A^2 -> H(U1(k, e), f(d))) (A^2 -> H(U1(c, e), U1(d, d)),A^2 -> H(U1(c, e), U1(d, d))) (A^2 -> H(U1(c, e), f(k)),A^2 -> H(U1(c, e), f(k))) ---------------------------------------- (378) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, k), U1(b, b)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (379) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(b, b)),A^2 -> H(U1(e, k), U1(b, b))) (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(U1(c, k), U1(c, b)),A^2 -> H(U1(c, k), U1(c, b))) (A^2 -> H(U1(c, k), U1(d, b)),A^2 -> H(U1(c, k), U1(d, b))) (A^2 -> H(U1(c, k), U1(b, c)),A^2 -> H(U1(c, k), U1(b, c))) (A^2 -> H(U1(c, k), U1(b, d)),A^2 -> H(U1(c, k), U1(b, d))) ---------------------------------------- (380) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, k), f(c)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (381) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), f(c)),A^2 -> H(U1(e, k), f(c))) (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(U1(c, k), U1(c, c)),A^2 -> H(U1(c, k), U1(c, c))) (A^2 -> H(U1(c, k), f(e)),A^2 -> H(U1(c, k), f(e))) (A^2 -> H(U1(c, k), f(k)),A^2 -> H(U1(c, k), f(k))) ---------------------------------------- (382) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, k), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (383) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), f(d)),A^2 -> H(U1(e, k), f(d))) (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) (A^2 -> H(U1(c, k), U1(d, d)),A^2 -> H(U1(c, k), U1(d, d))) (A^2 -> H(U1(c, k), f(k)),A^2 -> H(U1(c, k), f(k))) ---------------------------------------- (384) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, b)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (385) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(c, b)),A^2 -> H(U1(e, c), U1(c, b))) (A^2 -> H(U1(k, c), U1(c, b)),A^2 -> H(U1(k, c), U1(c, b))) (A^2 -> H(U1(c, e), U1(c, b)),A^2 -> H(U1(c, e), U1(c, b))) (A^2 -> H(U1(c, k), U1(c, b)),A^2 -> H(U1(c, k), U1(c, b))) (A^2 -> H(U1(c, c), U1(e, b)),A^2 -> H(U1(c, c), U1(e, b))) (A^2 -> H(U1(c, c), U1(k, b)),A^2 -> H(U1(c, c), U1(k, b))) (A^2 -> H(U1(c, c), U1(c, c)),A^2 -> H(U1(c, c), U1(c, c))) (A^2 -> H(U1(c, c), U1(c, d)),A^2 -> H(U1(c, c), U1(c, d))) ---------------------------------------- (386) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(d, b)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (387) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(d, b)),A^2 -> H(U1(e, c), U1(d, b))) (A^2 -> H(U1(k, c), U1(d, b)),A^2 -> H(U1(k, c), U1(d, b))) (A^2 -> H(U1(c, e), U1(d, b)),A^2 -> H(U1(c, e), U1(d, b))) (A^2 -> H(U1(c, k), U1(d, b)),A^2 -> H(U1(c, k), U1(d, b))) (A^2 -> H(U1(c, c), U1(k, b)),A^2 -> H(U1(c, c), U1(k, b))) (A^2 -> H(U1(c, c), U1(d, c)),A^2 -> H(U1(c, c), U1(d, c))) (A^2 -> H(U1(c, c), U1(d, d)),A^2 -> H(U1(c, c), U1(d, d))) ---------------------------------------- (388) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(b, c)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (389) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(b, c)),A^2 -> H(U1(e, c), U1(b, c))) (A^2 -> H(U1(k, c), U1(b, c)),A^2 -> H(U1(k, c), U1(b, c))) (A^2 -> H(U1(c, e), U1(b, c)),A^2 -> H(U1(c, e), U1(b, c))) (A^2 -> H(U1(c, k), U1(b, c)),A^2 -> H(U1(c, k), U1(b, c))) (A^2 -> H(U1(c, c), U1(c, c)),A^2 -> H(U1(c, c), U1(c, c))) (A^2 -> H(U1(c, c), U1(d, c)),A^2 -> H(U1(c, c), U1(d, c))) (A^2 -> H(U1(c, c), U1(b, e)),A^2 -> H(U1(c, c), U1(b, e))) (A^2 -> H(U1(c, c), U1(b, k)),A^2 -> H(U1(c, c), U1(b, k))) ---------------------------------------- (390) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(b, d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (391) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(b, d)),A^2 -> H(U1(e, c), U1(b, d))) (A^2 -> H(U1(k, c), U1(b, d)),A^2 -> H(U1(k, c), U1(b, d))) (A^2 -> H(U1(c, e), U1(b, d)),A^2 -> H(U1(c, e), U1(b, d))) (A^2 -> H(U1(c, k), U1(b, d)),A^2 -> H(U1(c, k), U1(b, d))) (A^2 -> H(U1(c, c), U1(c, d)),A^2 -> H(U1(c, c), U1(c, d))) (A^2 -> H(U1(c, c), U1(d, d)),A^2 -> H(U1(c, c), U1(d, d))) (A^2 -> H(U1(c, c), U1(b, k)),A^2 -> H(U1(c, c), U1(b, k))) ---------------------------------------- (392) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(e)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (393) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), f(e)),A^2 -> H(U1(e, c), f(e))) (A^2 -> H(U1(k, c), f(e)),A^2 -> H(U1(k, c), f(e))) (A^2 -> H(U1(c, e), f(e)),A^2 -> H(U1(c, e), f(e))) (A^2 -> H(U1(c, k), f(e)),A^2 -> H(U1(c, k), f(e))) (A^2 -> H(U1(c, c), U1(e, e)),A^2 -> H(U1(c, c), U1(e, e))) ---------------------------------------- (394) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), f(k)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (395) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), f(k)),A^2 -> H(U1(e, c), f(k))) (A^2 -> H(U1(k, c), f(k)),A^2 -> H(U1(k, c), f(k))) (A^2 -> H(U1(c, e), f(k)),A^2 -> H(U1(c, e), f(k))) (A^2 -> H(U1(c, k), f(k)),A^2 -> H(U1(c, k), f(k))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) ---------------------------------------- (396) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(c, c), U1(d, d)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (397) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(d, d)),A^2 -> H(U1(e, c), U1(d, d))) (A^2 -> H(U1(k, c), U1(d, d)),A^2 -> H(U1(k, c), U1(d, d))) (A^2 -> H(U1(c, e), U1(d, d)),A^2 -> H(U1(c, e), U1(d, d))) (A^2 -> H(U1(c, k), U1(d, d)),A^2 -> H(U1(c, k), U1(d, d))) (A^2 -> H(U1(c, c), U1(k, d)),A^2 -> H(U1(c, c), U1(k, d))) (A^2 -> H(U1(c, c), U1(d, k)),A^2 -> H(U1(c, c), U1(d, k))) ---------------------------------------- (398) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(e, f(b)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (399) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, b)),A^2 -> H(e, U1(b, b))) (A^2 -> H(e, f(c)),A^2 -> H(e, f(c))) (A^2 -> H(e, f(d)),A^2 -> H(e, f(d))) ---------------------------------------- (400) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(e, e), U1(b, b)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (401) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, b)),A^2 -> H(e, U1(b, b))) (A^2 -> H(U1(e, e), U1(c, b)),A^2 -> H(U1(e, e), U1(c, b))) (A^2 -> H(U1(e, e), U1(d, b)),A^2 -> H(U1(e, e), U1(d, b))) (A^2 -> H(U1(e, e), U1(b, c)),A^2 -> H(U1(e, e), U1(b, c))) (A^2 -> H(U1(e, e), U1(b, d)),A^2 -> H(U1(e, e), U1(b, d))) ---------------------------------------- (402) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(e, e), f(c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (403) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(c)),A^2 -> H(e, f(c))) (A^2 -> H(U1(e, e), U1(c, c)),A^2 -> H(U1(e, e), U1(c, c))) (A^2 -> H(U1(e, e), f(e)),A^2 -> H(U1(e, e), f(e))) (A^2 -> H(U1(e, e), f(k)),A^2 -> H(U1(e, e), f(k))) ---------------------------------------- (404) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(e, e), f(d)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (405) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(d)),A^2 -> H(e, f(d))) (A^2 -> H(U1(e, e), U1(d, d)),A^2 -> H(U1(e, e), U1(d, d))) (A^2 -> H(U1(e, e), f(k)),A^2 -> H(U1(e, e), f(k))) ---------------------------------------- (406) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), U1(c, b)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (407) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, b)),A^2 -> H(U1(e, e), U1(c, b))) (A^2 -> H(f(e), U1(e, b)),A^2 -> H(f(e), U1(e, b))) (A^2 -> H(f(e), U1(k, b)),A^2 -> H(f(e), U1(k, b))) (A^2 -> H(f(e), U1(c, c)),A^2 -> H(f(e), U1(c, c))) (A^2 -> H(f(e), U1(c, d)),A^2 -> H(f(e), U1(c, d))) ---------------------------------------- (408) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), U1(d, b)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (409) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, b)),A^2 -> H(U1(e, e), U1(d, b))) (A^2 -> H(f(e), U1(k, b)),A^2 -> H(f(e), U1(k, b))) (A^2 -> H(f(e), U1(d, c)),A^2 -> H(f(e), U1(d, c))) (A^2 -> H(f(e), U1(d, d)),A^2 -> H(f(e), U1(d, d))) ---------------------------------------- (410) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), U1(b, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (411) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, c)),A^2 -> H(U1(e, e), U1(b, c))) (A^2 -> H(f(e), U1(c, c)),A^2 -> H(f(e), U1(c, c))) (A^2 -> H(f(e), U1(d, c)),A^2 -> H(f(e), U1(d, c))) (A^2 -> H(f(e), U1(b, e)),A^2 -> H(f(e), U1(b, e))) (A^2 -> H(f(e), U1(b, k)),A^2 -> H(f(e), U1(b, k))) ---------------------------------------- (412) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), U1(b, d)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (413) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, d)),A^2 -> H(U1(e, e), U1(b, d))) (A^2 -> H(f(e), U1(c, d)),A^2 -> H(f(e), U1(c, d))) (A^2 -> H(f(e), U1(d, d)),A^2 -> H(f(e), U1(d, d))) (A^2 -> H(f(e), U1(b, k)),A^2 -> H(f(e), U1(b, k))) ---------------------------------------- (414) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (415) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, c)),A^2 -> H(U1(e, e), U1(c, c))) (A^2 -> H(f(e), U1(e, c)),A^2 -> H(f(e), U1(e, c))) (A^2 -> H(f(e), U1(k, c)),A^2 -> H(f(e), U1(k, c))) (A^2 -> H(f(e), U1(c, e)),A^2 -> H(f(e), U1(c, e))) (A^2 -> H(f(e), U1(c, k)),A^2 -> H(f(e), U1(c, k))) ---------------------------------------- (416) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), f(k)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (417) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(k)),A^2 -> H(U1(e, e), f(k))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) ---------------------------------------- (418) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(e), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (419) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, d)),A^2 -> H(U1(e, e), U1(d, d))) (A^2 -> H(f(e), U1(k, d)),A^2 -> H(f(e), U1(k, d))) (A^2 -> H(f(e), U1(d, k)),A^2 -> H(f(e), U1(d, k))) ---------------------------------------- (420) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (421) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) ---------------------------------------- (422) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (423) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) ---------------------------------------- (424) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (425) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) ---------------------------------------- (426) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), U1(c, b)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (427) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(f(k), U1(e, b)),A^2 -> H(f(k), U1(e, b))) (A^2 -> H(f(k), U1(k, b)),A^2 -> H(f(k), U1(k, b))) (A^2 -> H(f(k), U1(c, c)),A^2 -> H(f(k), U1(c, c))) (A^2 -> H(f(k), U1(c, d)),A^2 -> H(f(k), U1(c, d))) ---------------------------------------- (428) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), U1(d, b)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (429) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(f(k), U1(k, b)),A^2 -> H(f(k), U1(k, b))) (A^2 -> H(f(k), U1(d, c)),A^2 -> H(f(k), U1(d, c))) (A^2 -> H(f(k), U1(d, d)),A^2 -> H(f(k), U1(d, d))) ---------------------------------------- (430) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), U1(b, c)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (431) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(f(k), U1(c, c)),A^2 -> H(f(k), U1(c, c))) (A^2 -> H(f(k), U1(d, c)),A^2 -> H(f(k), U1(d, c))) (A^2 -> H(f(k), U1(b, e)),A^2 -> H(f(k), U1(b, e))) (A^2 -> H(f(k), U1(b, k)),A^2 -> H(f(k), U1(b, k))) ---------------------------------------- (432) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), U1(b, d)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (433) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) (A^2 -> H(f(k), U1(c, d)),A^2 -> H(f(k), U1(c, d))) (A^2 -> H(f(k), U1(d, d)),A^2 -> H(f(k), U1(d, d))) (A^2 -> H(f(k), U1(b, k)),A^2 -> H(f(k), U1(b, k))) ---------------------------------------- (434) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), U1(c, c)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (435) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(f(k), U1(e, c)),A^2 -> H(f(k), U1(e, c))) (A^2 -> H(f(k), U1(k, c)),A^2 -> H(f(k), U1(k, c))) (A^2 -> H(f(k), U1(c, e)),A^2 -> H(f(k), U1(c, e))) (A^2 -> H(f(k), U1(c, k)),A^2 -> H(f(k), U1(c, k))) ---------------------------------------- (436) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (437) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(f(k), U1(e, e)),A^2 -> H(f(k), U1(e, e))) ---------------------------------------- (438) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(k), U1(d, d)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (439) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(f(k), U1(k, d)),A^2 -> H(f(k), U1(k, d))) (A^2 -> H(f(k), U1(d, k)),A^2 -> H(f(k), U1(d, k))) ---------------------------------------- (440) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(e, b)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (441) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, b)),A^2 -> H(U1(c, c), U1(e, b))) (A^2 -> H(f(e), U1(e, b)),A^2 -> H(f(e), U1(e, b))) (A^2 -> H(f(k), U1(e, b)),A^2 -> H(f(k), U1(e, b))) (A^2 -> H(f(c), b),A^2 -> H(f(c), b)) (A^2 -> H(f(c), U1(e, c)),A^2 -> H(f(c), U1(e, c))) (A^2 -> H(f(c), U1(e, d)),A^2 -> H(f(c), U1(e, d))) ---------------------------------------- (442) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(k, b)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (443) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, b)),A^2 -> H(U1(c, c), U1(k, b))) (A^2 -> H(f(e), U1(k, b)),A^2 -> H(f(e), U1(k, b))) (A^2 -> H(f(k), U1(k, b)),A^2 -> H(f(k), U1(k, b))) (A^2 -> H(f(c), U1(k, c)),A^2 -> H(f(c), U1(k, c))) (A^2 -> H(f(c), U1(k, d)),A^2 -> H(f(c), U1(k, d))) ---------------------------------------- (444) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(c, d)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (445) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, d)),A^2 -> H(U1(c, c), U1(c, d))) (A^2 -> H(f(e), U1(c, d)),A^2 -> H(f(e), U1(c, d))) (A^2 -> H(f(k), U1(c, d)),A^2 -> H(f(k), U1(c, d))) (A^2 -> H(f(c), U1(e, d)),A^2 -> H(f(c), U1(e, d))) (A^2 -> H(f(c), U1(k, d)),A^2 -> H(f(c), U1(k, d))) (A^2 -> H(f(c), U1(c, k)),A^2 -> H(f(c), U1(c, k))) ---------------------------------------- (446) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(d, c)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (447) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, c)),A^2 -> H(U1(c, c), U1(d, c))) (A^2 -> H(f(e), U1(d, c)),A^2 -> H(f(e), U1(d, c))) (A^2 -> H(f(k), U1(d, c)),A^2 -> H(f(k), U1(d, c))) (A^2 -> H(f(c), U1(k, c)),A^2 -> H(f(c), U1(k, c))) (A^2 -> H(f(c), U1(d, e)),A^2 -> H(f(c), U1(d, e))) (A^2 -> H(f(c), U1(d, k)),A^2 -> H(f(c), U1(d, k))) ---------------------------------------- (448) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(b, e)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (449) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, e)),A^2 -> H(U1(c, c), U1(b, e))) (A^2 -> H(f(e), U1(b, e)),A^2 -> H(f(e), U1(b, e))) (A^2 -> H(f(k), U1(b, e)),A^2 -> H(f(k), U1(b, e))) (A^2 -> H(f(c), U1(c, e)),A^2 -> H(f(c), U1(c, e))) (A^2 -> H(f(c), U1(d, e)),A^2 -> H(f(c), U1(d, e))) ---------------------------------------- (450) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(b, k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (451) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, k)),A^2 -> H(U1(c, c), U1(b, k))) (A^2 -> H(f(e), U1(b, k)),A^2 -> H(f(e), U1(b, k))) (A^2 -> H(f(k), U1(b, k)),A^2 -> H(f(k), U1(b, k))) (A^2 -> H(f(c), U1(c, k)),A^2 -> H(f(c), U1(c, k))) (A^2 -> H(f(c), U1(d, k)),A^2 -> H(f(c), U1(d, k))) ---------------------------------------- (452) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(k, d)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (453) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, d)),A^2 -> H(U1(c, c), U1(k, d))) (A^2 -> H(f(e), U1(k, d)),A^2 -> H(f(e), U1(k, d))) (A^2 -> H(f(k), U1(k, d)),A^2 -> H(f(k), U1(k, d))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) ---------------------------------------- (454) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(d, k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (455) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, k)),A^2 -> H(U1(c, c), U1(d, k))) (A^2 -> H(f(e), U1(d, k)),A^2 -> H(f(e), U1(d, k))) (A^2 -> H(f(k), U1(d, k)),A^2 -> H(f(k), U1(d, k))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) ---------------------------------------- (456) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (457) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (458) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(k, k), f(b)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (459) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) ---------------------------------------- (460) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(k, d), U1(b, b)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (461) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(U1(k, d), U1(c, b)),A^2 -> H(U1(k, d), U1(c, b))) (A^2 -> H(U1(k, d), U1(d, b)),A^2 -> H(U1(k, d), U1(d, b))) (A^2 -> H(U1(k, d), U1(b, c)),A^2 -> H(U1(k, d), U1(b, c))) (A^2 -> H(U1(k, d), U1(b, d)),A^2 -> H(U1(k, d), U1(b, d))) ---------------------------------------- (462) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(k, d), f(c)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (463) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(U1(k, d), U1(c, c)),A^2 -> H(U1(k, d), U1(c, c))) (A^2 -> H(U1(k, d), f(e)),A^2 -> H(U1(k, d), f(e))) (A^2 -> H(U1(k, d), f(k)),A^2 -> H(U1(k, d), f(k))) ---------------------------------------- (464) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(k, d), f(d)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (465) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) (A^2 -> H(U1(k, d), U1(d, d)),A^2 -> H(U1(k, d), U1(d, d))) (A^2 -> H(U1(k, d), f(k)),A^2 -> H(U1(k, d), f(k))) ---------------------------------------- (466) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, k), U1(b, b)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (467) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, b)),A^2 -> H(U1(k, k), U1(b, b))) (A^2 -> H(U1(d, k), U1(c, b)),A^2 -> H(U1(d, k), U1(c, b))) (A^2 -> H(U1(d, k), U1(d, b)),A^2 -> H(U1(d, k), U1(d, b))) (A^2 -> H(U1(d, k), U1(b, c)),A^2 -> H(U1(d, k), U1(b, c))) (A^2 -> H(U1(d, k), U1(b, d)),A^2 -> H(U1(d, k), U1(b, d))) ---------------------------------------- (468) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, k), f(c)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (469) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(c)),A^2 -> H(U1(k, k), f(c))) (A^2 -> H(U1(d, k), U1(c, c)),A^2 -> H(U1(d, k), U1(c, c))) (A^2 -> H(U1(d, k), f(e)),A^2 -> H(U1(d, k), f(e))) (A^2 -> H(U1(d, k), f(k)),A^2 -> H(U1(d, k), f(k))) ---------------------------------------- (470) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, k), f(d)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (471) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(d)),A^2 -> H(U1(k, k), f(d))) (A^2 -> H(U1(d, k), U1(d, d)),A^2 -> H(U1(d, k), U1(d, d))) (A^2 -> H(U1(d, k), f(k)),A^2 -> H(U1(d, k), f(k))) ---------------------------------------- (472) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(c, b)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (473) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(c, b)),A^2 -> H(U1(k, d), U1(c, b))) (A^2 -> H(U1(d, k), U1(c, b)),A^2 -> H(U1(d, k), U1(c, b))) (A^2 -> H(U1(d, d), U1(e, b)),A^2 -> H(U1(d, d), U1(e, b))) (A^2 -> H(U1(d, d), U1(k, b)),A^2 -> H(U1(d, d), U1(k, b))) (A^2 -> H(U1(d, d), U1(c, c)),A^2 -> H(U1(d, d), U1(c, c))) (A^2 -> H(U1(d, d), U1(c, d)),A^2 -> H(U1(d, d), U1(c, d))) ---------------------------------------- (474) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, b)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (475) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(d, b)),A^2 -> H(U1(k, d), U1(d, b))) (A^2 -> H(U1(d, k), U1(d, b)),A^2 -> H(U1(d, k), U1(d, b))) (A^2 -> H(U1(d, d), U1(k, b)),A^2 -> H(U1(d, d), U1(k, b))) (A^2 -> H(U1(d, d), U1(d, c)),A^2 -> H(U1(d, d), U1(d, c))) (A^2 -> H(U1(d, d), U1(d, d)),A^2 -> H(U1(d, d), U1(d, d))) ---------------------------------------- (476) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(b, c)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (477) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(b, c)),A^2 -> H(U1(k, d), U1(b, c))) (A^2 -> H(U1(d, k), U1(b, c)),A^2 -> H(U1(d, k), U1(b, c))) (A^2 -> H(U1(d, d), U1(c, c)),A^2 -> H(U1(d, d), U1(c, c))) (A^2 -> H(U1(d, d), U1(d, c)),A^2 -> H(U1(d, d), U1(d, c))) (A^2 -> H(U1(d, d), U1(b, e)),A^2 -> H(U1(d, d), U1(b, e))) (A^2 -> H(U1(d, d), U1(b, k)),A^2 -> H(U1(d, d), U1(b, k))) ---------------------------------------- (478) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(b, d)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (479) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(b, d)),A^2 -> H(U1(k, d), U1(b, d))) (A^2 -> H(U1(d, k), U1(b, d)),A^2 -> H(U1(d, k), U1(b, d))) (A^2 -> H(U1(d, d), U1(c, d)),A^2 -> H(U1(d, d), U1(c, d))) (A^2 -> H(U1(d, d), U1(d, d)),A^2 -> H(U1(d, d), U1(d, d))) (A^2 -> H(U1(d, d), U1(b, k)),A^2 -> H(U1(d, d), U1(b, k))) ---------------------------------------- (480) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(c, c)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (481) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(c, c)),A^2 -> H(U1(k, d), U1(c, c))) (A^2 -> H(U1(d, k), U1(c, c)),A^2 -> H(U1(d, k), U1(c, c))) (A^2 -> H(U1(d, d), U1(e, c)),A^2 -> H(U1(d, d), U1(e, c))) (A^2 -> H(U1(d, d), U1(k, c)),A^2 -> H(U1(d, d), U1(k, c))) (A^2 -> H(U1(d, d), U1(c, e)),A^2 -> H(U1(d, d), U1(c, e))) (A^2 -> H(U1(d, d), U1(c, k)),A^2 -> H(U1(d, d), U1(c, k))) ---------------------------------------- (482) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), f(e)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (483) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), f(e)),A^2 -> H(U1(k, d), f(e))) (A^2 -> H(U1(d, k), f(e)),A^2 -> H(U1(d, k), f(e))) (A^2 -> H(U1(d, d), U1(e, e)),A^2 -> H(U1(d, d), U1(e, e))) ---------------------------------------- (484) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (485) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), f(k)),A^2 -> H(U1(k, d), f(k))) (A^2 -> H(U1(d, k), f(k)),A^2 -> H(U1(d, k), f(k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (486) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, b)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (487) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(e, b)),A^2 -> H(U1(d, d), U1(e, b))) (A^2 -> H(f(k), U1(e, b)),A^2 -> H(f(k), U1(e, b))) (A^2 -> H(f(d), b),A^2 -> H(f(d), b)) (A^2 -> H(f(d), U1(e, c)),A^2 -> H(f(d), U1(e, c))) (A^2 -> H(f(d), U1(e, d)),A^2 -> H(f(d), U1(e, d))) ---------------------------------------- (488) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(k, b)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (489) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(k, b)),A^2 -> H(U1(d, d), U1(k, b))) (A^2 -> H(f(k), U1(k, b)),A^2 -> H(f(k), U1(k, b))) (A^2 -> H(f(d), U1(k, c)),A^2 -> H(f(d), U1(k, c))) (A^2 -> H(f(d), U1(k, d)),A^2 -> H(f(d), U1(k, d))) ---------------------------------------- (490) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(c, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (491) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(c, d)),A^2 -> H(U1(d, d), U1(c, d))) (A^2 -> H(f(k), U1(c, d)),A^2 -> H(f(k), U1(c, d))) (A^2 -> H(f(d), U1(e, d)),A^2 -> H(f(d), U1(e, d))) (A^2 -> H(f(d), U1(k, d)),A^2 -> H(f(d), U1(k, d))) (A^2 -> H(f(d), U1(c, k)),A^2 -> H(f(d), U1(c, k))) ---------------------------------------- (492) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(d, c)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (493) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(d, c)),A^2 -> H(U1(d, d), U1(d, c))) (A^2 -> H(f(k), U1(d, c)),A^2 -> H(f(k), U1(d, c))) (A^2 -> H(f(d), U1(k, c)),A^2 -> H(f(d), U1(k, c))) (A^2 -> H(f(d), U1(d, e)),A^2 -> H(f(d), U1(d, e))) (A^2 -> H(f(d), U1(d, k)),A^2 -> H(f(d), U1(d, k))) ---------------------------------------- (494) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(b, e)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (495) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(b, e)),A^2 -> H(U1(d, d), U1(b, e))) (A^2 -> H(f(k), U1(b, e)),A^2 -> H(f(k), U1(b, e))) (A^2 -> H(f(d), U1(c, e)),A^2 -> H(f(d), U1(c, e))) (A^2 -> H(f(d), U1(d, e)),A^2 -> H(f(d), U1(d, e))) ---------------------------------------- (496) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(b, k)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (497) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(b, k)),A^2 -> H(U1(d, d), U1(b, k))) (A^2 -> H(f(k), U1(b, k)),A^2 -> H(f(k), U1(b, k))) (A^2 -> H(f(d), U1(c, k)),A^2 -> H(f(d), U1(c, k))) (A^2 -> H(f(d), U1(d, k)),A^2 -> H(f(d), U1(d, k))) ---------------------------------------- (498) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, c)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (499) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(e, c)),A^2 -> H(U1(d, d), U1(e, c))) (A^2 -> H(f(k), U1(e, c)),A^2 -> H(f(k), U1(e, c))) (A^2 -> H(f(d), c),A^2 -> H(f(d), c)) (A^2 -> H(f(d), U1(e, e)),A^2 -> H(f(d), U1(e, e))) (A^2 -> H(f(d), U1(e, k)),A^2 -> H(f(d), U1(e, k))) ---------------------------------------- (500) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(k, c)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (501) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(k, c)),A^2 -> H(U1(d, d), U1(k, c))) (A^2 -> H(f(k), U1(k, c)),A^2 -> H(f(k), U1(k, c))) (A^2 -> H(f(d), U1(k, e)),A^2 -> H(f(d), U1(k, e))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (502) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(c, e)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (503) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(c, e)),A^2 -> H(U1(d, d), U1(c, e))) (A^2 -> H(f(k), U1(c, e)),A^2 -> H(f(k), U1(c, e))) (A^2 -> H(f(d), U1(e, e)),A^2 -> H(f(d), U1(e, e))) (A^2 -> H(f(d), U1(k, e)),A^2 -> H(f(d), U1(k, e))) ---------------------------------------- (504) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(c, k)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (505) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(c, k)),A^2 -> H(U1(d, d), U1(c, k))) (A^2 -> H(f(k), U1(c, k)),A^2 -> H(f(k), U1(c, k))) (A^2 -> H(f(d), U1(e, k)),A^2 -> H(f(d), U1(e, k))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (506) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, e)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (507) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(e, e)),A^2 -> H(U1(d, d), U1(e, e))) (A^2 -> H(f(k), U1(e, e)),A^2 -> H(f(k), U1(e, e))) (A^2 -> H(f(d), e),A^2 -> H(f(d), e)) ---------------------------------------- (508) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (509) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (510) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), b) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (511) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), b),A^2 -> H(U1(a, a), b)) (A^2 -> H(f(c), b),A^2 -> H(f(c), b)) (A^2 -> H(f(d), b),A^2 -> H(f(d), b)) (A^2 -> H(f(a), c),A^2 -> H(f(a), c)) (A^2 -> H(f(a), d),A^2 -> H(f(a), d)) ---------------------------------------- (512) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), U1(e, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (513) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(e, d)),A^2 -> H(U1(a, a), U1(e, d))) (A^2 -> H(f(c), U1(e, d)),A^2 -> H(f(c), U1(e, d))) (A^2 -> H(f(d), U1(e, d)),A^2 -> H(f(d), U1(e, d))) (A^2 -> H(f(a), d),A^2 -> H(f(a), d)) (A^2 -> H(f(a), U1(e, k)),A^2 -> H(f(a), U1(e, k))) ---------------------------------------- (514) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), U1(d, e)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (515) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(d, e)),A^2 -> H(U1(a, a), U1(d, e))) (A^2 -> H(f(c), U1(d, e)),A^2 -> H(f(c), U1(d, e))) (A^2 -> H(f(d), U1(d, e)),A^2 -> H(f(d), U1(d, e))) (A^2 -> H(f(a), U1(k, e)),A^2 -> H(f(a), U1(k, e))) ---------------------------------------- (516) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(e, c)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (517) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, c)),A^2 -> H(U1(c, c), U1(e, c))) (A^2 -> H(f(e), U1(e, c)),A^2 -> H(f(e), U1(e, c))) (A^2 -> H(f(k), U1(e, c)),A^2 -> H(f(k), U1(e, c))) (A^2 -> H(f(c), c),A^2 -> H(f(c), c)) (A^2 -> H(f(c), U1(e, e)),A^2 -> H(f(c), U1(e, e))) (A^2 -> H(f(c), U1(e, k)),A^2 -> H(f(c), U1(e, k))) ---------------------------------------- (518) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(k, c)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (519) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, c)),A^2 -> H(U1(c, c), U1(k, c))) (A^2 -> H(f(e), U1(k, c)),A^2 -> H(f(e), U1(k, c))) (A^2 -> H(f(k), U1(k, c)),A^2 -> H(f(k), U1(k, c))) (A^2 -> H(f(c), U1(k, e)),A^2 -> H(f(c), U1(k, e))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) ---------------------------------------- (520) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(c, e)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (521) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, e)),A^2 -> H(U1(c, c), U1(c, e))) (A^2 -> H(f(e), U1(c, e)),A^2 -> H(f(e), U1(c, e))) (A^2 -> H(f(k), U1(c, e)),A^2 -> H(f(k), U1(c, e))) (A^2 -> H(f(c), U1(e, e)),A^2 -> H(f(c), U1(e, e))) (A^2 -> H(f(c), U1(k, e)),A^2 -> H(f(c), U1(k, e))) ---------------------------------------- (522) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(c, k)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (523) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, k)),A^2 -> H(U1(c, c), U1(c, k))) (A^2 -> H(f(e), U1(c, k)),A^2 -> H(f(e), U1(c, k))) (A^2 -> H(f(k), U1(c, k)),A^2 -> H(f(k), U1(c, k))) (A^2 -> H(f(c), U1(e, k)),A^2 -> H(f(c), U1(e, k))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) ---------------------------------------- (524) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), c) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (525) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), c),A^2 -> H(U1(a, a), c)) (A^2 -> H(f(c), c),A^2 -> H(f(c), c)) (A^2 -> H(f(d), c),A^2 -> H(f(d), c)) (A^2 -> H(f(a), e),A^2 -> H(f(a), e)) (A^2 -> H(f(a), k),A^2 -> H(f(a), k)) ---------------------------------------- (526) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), U1(e, k)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (527) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(e, k)),A^2 -> H(U1(a, a), U1(e, k))) (A^2 -> H(f(c), U1(e, k)),A^2 -> H(f(c), U1(e, k))) (A^2 -> H(f(d), U1(e, k)),A^2 -> H(f(d), U1(e, k))) (A^2 -> H(f(a), k),A^2 -> H(f(a), k)) ---------------------------------------- (528) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), U1(k, e)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (529) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(k, e)),A^2 -> H(U1(a, a), U1(k, e))) (A^2 -> H(f(c), U1(k, e)),A^2 -> H(f(c), U1(k, e))) (A^2 -> H(f(d), U1(k, e)),A^2 -> H(f(d), U1(k, e))) ---------------------------------------- (530) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(e, e)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (531) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, e)),A^2 -> H(U1(c, c), U1(e, e))) (A^2 -> H(f(e), U1(e, e)),A^2 -> H(f(e), U1(e, e))) (A^2 -> H(f(k), U1(e, e)),A^2 -> H(f(k), U1(e, e))) (A^2 -> H(f(c), e),A^2 -> H(f(c), e)) ---------------------------------------- (532) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), e) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (533) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), e),A^2 -> H(U1(a, a), e)) (A^2 -> H(f(c), e),A^2 -> H(f(c), e)) (A^2 -> H(f(d), e),A^2 -> H(f(d), e)) ---------------------------------------- (534) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(k, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (535) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(k, d)),A^2 -> H(U1(d, d), U1(k, d))) (A^2 -> H(f(k), U1(k, d)),A^2 -> H(f(k), U1(k, d))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (536) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(d, k)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (537) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(d, k)),A^2 -> H(U1(d, d), U1(d, k))) (A^2 -> H(f(k), U1(d, k)),A^2 -> H(f(k), U1(d, k))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (538) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), U1(k, k)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (539) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), U1(k, k)),A^2 -> H(U1(a, a), U1(k, k))) (A^2 -> H(f(c), U1(k, k)),A^2 -> H(f(c), U1(k, k))) (A^2 -> H(f(d), U1(k, k)),A^2 -> H(f(d), U1(k, k))) ---------------------------------------- (540) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(d, f(b)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (541) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(b)),A^2 -> H(k, f(b))) (A^2 -> H(d, U1(b, b)),A^2 -> H(d, U1(b, b))) (A^2 -> H(d, f(c)),A^2 -> H(d, f(c))) (A^2 -> H(d, f(d)),A^2 -> H(d, f(d))) ---------------------------------------- (542) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(b, b)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (543) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, b)),A^2 -> H(c, U1(b, b))) (A^2 -> H(d, U1(b, b)),A^2 -> H(d, U1(b, b))) (A^2 -> H(a, U1(c, b)),A^2 -> H(a, U1(c, b))) (A^2 -> H(a, U1(d, b)),A^2 -> H(a, U1(d, b))) (A^2 -> H(a, U1(b, c)),A^2 -> H(a, U1(b, c))) (A^2 -> H(a, U1(b, d)),A^2 -> H(a, U1(b, d))) ---------------------------------------- (544) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, f(c)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (545) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(c)),A^2 -> H(c, f(c))) (A^2 -> H(d, f(c)),A^2 -> H(d, f(c))) (A^2 -> H(a, U1(c, c)),A^2 -> H(a, U1(c, c))) (A^2 -> H(a, f(e)),A^2 -> H(a, f(e))) (A^2 -> H(a, f(k)),A^2 -> H(a, f(k))) ---------------------------------------- (546) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, f(d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (547) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(d)),A^2 -> H(c, f(d))) (A^2 -> H(d, f(d)),A^2 -> H(d, f(d))) (A^2 -> H(a, U1(d, d)),A^2 -> H(a, U1(d, d))) (A^2 -> H(a, f(k)),A^2 -> H(a, f(k))) ---------------------------------------- (548) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(b, b)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (549) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(b, b)),A^2 -> H(d, U1(b, b))) (A^2 -> H(U1(e, k), U1(b, b)),A^2 -> H(U1(e, k), U1(b, b))) (A^2 -> H(U1(e, d), U1(c, b)),A^2 -> H(U1(e, d), U1(c, b))) (A^2 -> H(U1(e, d), U1(d, b)),A^2 -> H(U1(e, d), U1(d, b))) (A^2 -> H(U1(e, d), U1(b, c)),A^2 -> H(U1(e, d), U1(b, c))) (A^2 -> H(U1(e, d), U1(b, d)),A^2 -> H(U1(e, d), U1(b, d))) ---------------------------------------- (550) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), f(c)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (551) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, f(c)),A^2 -> H(d, f(c))) (A^2 -> H(U1(e, k), f(c)),A^2 -> H(U1(e, k), f(c))) (A^2 -> H(U1(e, d), U1(c, c)),A^2 -> H(U1(e, d), U1(c, c))) (A^2 -> H(U1(e, d), f(e)),A^2 -> H(U1(e, d), f(e))) (A^2 -> H(U1(e, d), f(k)),A^2 -> H(U1(e, d), f(k))) ---------------------------------------- (552) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), f(d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (553) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, f(d)),A^2 -> H(d, f(d))) (A^2 -> H(U1(e, k), f(d)),A^2 -> H(U1(e, k), f(d))) (A^2 -> H(U1(e, d), U1(d, d)),A^2 -> H(U1(e, d), U1(d, d))) (A^2 -> H(U1(e, d), f(k)),A^2 -> H(U1(e, d), f(k))) ---------------------------------------- (554) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(c, b)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (555) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(c, b)),A^2 -> H(a, U1(c, b))) (A^2 -> H(U1(e, c), U1(c, b)),A^2 -> H(U1(e, c), U1(c, b))) (A^2 -> H(U1(e, d), U1(c, b)),A^2 -> H(U1(e, d), U1(c, b))) (A^2 -> H(U1(e, a), U1(e, b)),A^2 -> H(U1(e, a), U1(e, b))) (A^2 -> H(U1(e, a), U1(k, b)),A^2 -> H(U1(e, a), U1(k, b))) (A^2 -> H(U1(e, a), U1(c, c)),A^2 -> H(U1(e, a), U1(c, c))) (A^2 -> H(U1(e, a), U1(c, d)),A^2 -> H(U1(e, a), U1(c, d))) ---------------------------------------- (556) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(d, b)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (557) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(d, b)),A^2 -> H(a, U1(d, b))) (A^2 -> H(U1(e, c), U1(d, b)),A^2 -> H(U1(e, c), U1(d, b))) (A^2 -> H(U1(e, d), U1(d, b)),A^2 -> H(U1(e, d), U1(d, b))) (A^2 -> H(U1(e, a), U1(k, b)),A^2 -> H(U1(e, a), U1(k, b))) (A^2 -> H(U1(e, a), U1(d, c)),A^2 -> H(U1(e, a), U1(d, c))) (A^2 -> H(U1(e, a), U1(d, d)),A^2 -> H(U1(e, a), U1(d, d))) ---------------------------------------- (558) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(b, c)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (559) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(b, c)),A^2 -> H(a, U1(b, c))) (A^2 -> H(U1(e, c), U1(b, c)),A^2 -> H(U1(e, c), U1(b, c))) (A^2 -> H(U1(e, d), U1(b, c)),A^2 -> H(U1(e, d), U1(b, c))) (A^2 -> H(U1(e, a), U1(c, c)),A^2 -> H(U1(e, a), U1(c, c))) (A^2 -> H(U1(e, a), U1(d, c)),A^2 -> H(U1(e, a), U1(d, c))) (A^2 -> H(U1(e, a), U1(b, e)),A^2 -> H(U1(e, a), U1(b, e))) (A^2 -> H(U1(e, a), U1(b, k)),A^2 -> H(U1(e, a), U1(b, k))) ---------------------------------------- (560) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(b, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (561) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(b, d)),A^2 -> H(a, U1(b, d))) (A^2 -> H(U1(e, c), U1(b, d)),A^2 -> H(U1(e, c), U1(b, d))) (A^2 -> H(U1(e, d), U1(b, d)),A^2 -> H(U1(e, d), U1(b, d))) (A^2 -> H(U1(e, a), U1(c, d)),A^2 -> H(U1(e, a), U1(c, d))) (A^2 -> H(U1(e, a), U1(d, d)),A^2 -> H(U1(e, a), U1(d, d))) (A^2 -> H(U1(e, a), U1(b, k)),A^2 -> H(U1(e, a), U1(b, k))) ---------------------------------------- (562) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(c, c)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (563) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(c, c)),A^2 -> H(a, U1(c, c))) (A^2 -> H(U1(e, c), U1(c, c)),A^2 -> H(U1(e, c), U1(c, c))) (A^2 -> H(U1(e, d), U1(c, c)),A^2 -> H(U1(e, d), U1(c, c))) (A^2 -> H(U1(e, a), U1(e, c)),A^2 -> H(U1(e, a), U1(e, c))) (A^2 -> H(U1(e, a), U1(k, c)),A^2 -> H(U1(e, a), U1(k, c))) (A^2 -> H(U1(e, a), U1(c, e)),A^2 -> H(U1(e, a), U1(c, e))) (A^2 -> H(U1(e, a), U1(c, k)),A^2 -> H(U1(e, a), U1(c, k))) ---------------------------------------- (564) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), f(e)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (565) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, f(e)),A^2 -> H(a, f(e))) (A^2 -> H(U1(e, c), f(e)),A^2 -> H(U1(e, c), f(e))) (A^2 -> H(U1(e, d), f(e)),A^2 -> H(U1(e, d), f(e))) (A^2 -> H(U1(e, a), U1(e, e)),A^2 -> H(U1(e, a), U1(e, e))) ---------------------------------------- (566) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), f(k)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (567) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, f(k)),A^2 -> H(a, f(k))) (A^2 -> H(U1(e, c), f(k)),A^2 -> H(U1(e, c), f(k))) (A^2 -> H(U1(e, d), f(k)),A^2 -> H(U1(e, d), f(k))) (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) ---------------------------------------- (568) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (569) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(d, d)),A^2 -> H(a, U1(d, d))) (A^2 -> H(U1(e, c), U1(d, d)),A^2 -> H(U1(e, c), U1(d, d))) (A^2 -> H(U1(e, d), U1(d, d)),A^2 -> H(U1(e, d), U1(d, d))) (A^2 -> H(U1(e, a), U1(k, d)),A^2 -> H(U1(e, a), U1(k, d))) (A^2 -> H(U1(e, a), U1(d, k)),A^2 -> H(U1(e, a), U1(d, k))) ---------------------------------------- (570) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(c, b)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (571) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, b)),A^2 -> H(U1(k, c), U1(c, b))) (A^2 -> H(U1(k, d), U1(c, b)),A^2 -> H(U1(k, d), U1(c, b))) (A^2 -> H(U1(k, a), U1(e, b)),A^2 -> H(U1(k, a), U1(e, b))) (A^2 -> H(U1(k, a), U1(k, b)),A^2 -> H(U1(k, a), U1(k, b))) (A^2 -> H(U1(k, a), U1(c, c)),A^2 -> H(U1(k, a), U1(c, c))) (A^2 -> H(U1(k, a), U1(c, d)),A^2 -> H(U1(k, a), U1(c, d))) ---------------------------------------- (572) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(d, b)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (573) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, b)),A^2 -> H(U1(k, c), U1(d, b))) (A^2 -> H(U1(k, d), U1(d, b)),A^2 -> H(U1(k, d), U1(d, b))) (A^2 -> H(U1(k, a), U1(k, b)),A^2 -> H(U1(k, a), U1(k, b))) (A^2 -> H(U1(k, a), U1(d, c)),A^2 -> H(U1(k, a), U1(d, c))) (A^2 -> H(U1(k, a), U1(d, d)),A^2 -> H(U1(k, a), U1(d, d))) ---------------------------------------- (574) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(b, c)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (575) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, c)),A^2 -> H(U1(k, c), U1(b, c))) (A^2 -> H(U1(k, d), U1(b, c)),A^2 -> H(U1(k, d), U1(b, c))) (A^2 -> H(U1(k, a), U1(c, c)),A^2 -> H(U1(k, a), U1(c, c))) (A^2 -> H(U1(k, a), U1(d, c)),A^2 -> H(U1(k, a), U1(d, c))) (A^2 -> H(U1(k, a), U1(b, e)),A^2 -> H(U1(k, a), U1(b, e))) (A^2 -> H(U1(k, a), U1(b, k)),A^2 -> H(U1(k, a), U1(b, k))) ---------------------------------------- (576) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(b, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (577) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, d)),A^2 -> H(U1(k, c), U1(b, d))) (A^2 -> H(U1(k, d), U1(b, d)),A^2 -> H(U1(k, d), U1(b, d))) (A^2 -> H(U1(k, a), U1(c, d)),A^2 -> H(U1(k, a), U1(c, d))) (A^2 -> H(U1(k, a), U1(d, d)),A^2 -> H(U1(k, a), U1(d, d))) (A^2 -> H(U1(k, a), U1(b, k)),A^2 -> H(U1(k, a), U1(b, k))) ---------------------------------------- (578) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(c, c)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (579) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, c)),A^2 -> H(U1(k, c), U1(c, c))) (A^2 -> H(U1(k, d), U1(c, c)),A^2 -> H(U1(k, d), U1(c, c))) (A^2 -> H(U1(k, a), U1(e, c)),A^2 -> H(U1(k, a), U1(e, c))) (A^2 -> H(U1(k, a), U1(k, c)),A^2 -> H(U1(k, a), U1(k, c))) (A^2 -> H(U1(k, a), U1(c, e)),A^2 -> H(U1(k, a), U1(c, e))) (A^2 -> H(U1(k, a), U1(c, k)),A^2 -> H(U1(k, a), U1(c, k))) ---------------------------------------- (580) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), f(e)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (581) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(e)),A^2 -> H(U1(k, c), f(e))) (A^2 -> H(U1(k, d), f(e)),A^2 -> H(U1(k, d), f(e))) (A^2 -> H(U1(k, a), U1(e, e)),A^2 -> H(U1(k, a), U1(e, e))) ---------------------------------------- (582) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), f(k)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (583) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(k)),A^2 -> H(U1(k, c), f(k))) (A^2 -> H(U1(k, d), f(k)),A^2 -> H(U1(k, d), f(k))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) ---------------------------------------- (584) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (585) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, d)),A^2 -> H(U1(k, c), U1(d, d))) (A^2 -> H(U1(k, d), U1(d, d)),A^2 -> H(U1(k, d), U1(d, d))) (A^2 -> H(U1(k, a), U1(k, d)),A^2 -> H(U1(k, a), U1(k, d))) (A^2 -> H(U1(k, a), U1(d, k)),A^2 -> H(U1(k, a), U1(d, k))) ---------------------------------------- (586) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, b)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (587) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(c, b)),A^2 -> H(U1(e, d), U1(c, b))) (A^2 -> H(U1(k, d), U1(c, b)),A^2 -> H(U1(k, d), U1(c, b))) (A^2 -> H(U1(c, k), U1(c, b)),A^2 -> H(U1(c, k), U1(c, b))) (A^2 -> H(U1(c, d), U1(e, b)),A^2 -> H(U1(c, d), U1(e, b))) (A^2 -> H(U1(c, d), U1(k, b)),A^2 -> H(U1(c, d), U1(k, b))) (A^2 -> H(U1(c, d), U1(c, c)),A^2 -> H(U1(c, d), U1(c, c))) (A^2 -> H(U1(c, d), U1(c, d)),A^2 -> H(U1(c, d), U1(c, d))) ---------------------------------------- (588) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(d, b)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (589) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(d, b)),A^2 -> H(U1(e, d), U1(d, b))) (A^2 -> H(U1(k, d), U1(d, b)),A^2 -> H(U1(k, d), U1(d, b))) (A^2 -> H(U1(c, k), U1(d, b)),A^2 -> H(U1(c, k), U1(d, b))) (A^2 -> H(U1(c, d), U1(k, b)),A^2 -> H(U1(c, d), U1(k, b))) (A^2 -> H(U1(c, d), U1(d, c)),A^2 -> H(U1(c, d), U1(d, c))) (A^2 -> H(U1(c, d), U1(d, d)),A^2 -> H(U1(c, d), U1(d, d))) ---------------------------------------- (590) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(b, c)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (591) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(b, c)),A^2 -> H(U1(e, d), U1(b, c))) (A^2 -> H(U1(k, d), U1(b, c)),A^2 -> H(U1(k, d), U1(b, c))) (A^2 -> H(U1(c, k), U1(b, c)),A^2 -> H(U1(c, k), U1(b, c))) (A^2 -> H(U1(c, d), U1(c, c)),A^2 -> H(U1(c, d), U1(c, c))) (A^2 -> H(U1(c, d), U1(d, c)),A^2 -> H(U1(c, d), U1(d, c))) (A^2 -> H(U1(c, d), U1(b, e)),A^2 -> H(U1(c, d), U1(b, e))) (A^2 -> H(U1(c, d), U1(b, k)),A^2 -> H(U1(c, d), U1(b, k))) ---------------------------------------- (592) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(b, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (593) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(b, d)),A^2 -> H(U1(e, d), U1(b, d))) (A^2 -> H(U1(k, d), U1(b, d)),A^2 -> H(U1(k, d), U1(b, d))) (A^2 -> H(U1(c, k), U1(b, d)),A^2 -> H(U1(c, k), U1(b, d))) (A^2 -> H(U1(c, d), U1(c, d)),A^2 -> H(U1(c, d), U1(c, d))) (A^2 -> H(U1(c, d), U1(d, d)),A^2 -> H(U1(c, d), U1(d, d))) (A^2 -> H(U1(c, d), U1(b, k)),A^2 -> H(U1(c, d), U1(b, k))) ---------------------------------------- (594) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, c)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (595) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(c, c)),A^2 -> H(U1(e, d), U1(c, c))) (A^2 -> H(U1(k, d), U1(c, c)),A^2 -> H(U1(k, d), U1(c, c))) (A^2 -> H(U1(c, k), U1(c, c)),A^2 -> H(U1(c, k), U1(c, c))) (A^2 -> H(U1(c, d), U1(e, c)),A^2 -> H(U1(c, d), U1(e, c))) (A^2 -> H(U1(c, d), U1(k, c)),A^2 -> H(U1(c, d), U1(k, c))) (A^2 -> H(U1(c, d), U1(c, e)),A^2 -> H(U1(c, d), U1(c, e))) (A^2 -> H(U1(c, d), U1(c, k)),A^2 -> H(U1(c, d), U1(c, k))) ---------------------------------------- (596) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), f(e)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (597) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), f(e)),A^2 -> H(U1(e, d), f(e))) (A^2 -> H(U1(k, d), f(e)),A^2 -> H(U1(k, d), f(e))) (A^2 -> H(U1(c, k), f(e)),A^2 -> H(U1(c, k), f(e))) (A^2 -> H(U1(c, d), U1(e, e)),A^2 -> H(U1(c, d), U1(e, e))) ---------------------------------------- (598) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), f(k)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (599) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), f(k)),A^2 -> H(U1(e, d), f(k))) (A^2 -> H(U1(k, d), f(k)),A^2 -> H(U1(k, d), f(k))) (A^2 -> H(U1(c, k), f(k)),A^2 -> H(U1(c, k), f(k))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (600) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (601) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(d, d)),A^2 -> H(U1(e, d), U1(d, d))) (A^2 -> H(U1(k, d), U1(d, d)),A^2 -> H(U1(k, d), U1(d, d))) (A^2 -> H(U1(c, k), U1(d, d)),A^2 -> H(U1(c, k), U1(d, d))) (A^2 -> H(U1(c, d), U1(k, d)),A^2 -> H(U1(c, d), U1(k, d))) (A^2 -> H(U1(c, d), U1(d, k)),A^2 -> H(U1(c, d), U1(d, k))) ---------------------------------------- (602) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, b)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (603) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(e, b)),A^2 -> H(U1(e, a), U1(e, b))) (A^2 -> H(U1(k, a), U1(e, b)),A^2 -> H(U1(k, a), U1(e, b))) (A^2 -> H(U1(c, c), U1(e, b)),A^2 -> H(U1(c, c), U1(e, b))) (A^2 -> H(U1(c, d), U1(e, b)),A^2 -> H(U1(c, d), U1(e, b))) (A^2 -> H(U1(c, a), b),A^2 -> H(U1(c, a), b)) (A^2 -> H(U1(c, a), U1(e, c)),A^2 -> H(U1(c, a), U1(e, c))) (A^2 -> H(U1(c, a), U1(e, d)),A^2 -> H(U1(c, a), U1(e, d))) ---------------------------------------- (604) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(k, b)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (605) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(k, b)),A^2 -> H(U1(e, a), U1(k, b))) (A^2 -> H(U1(k, a), U1(k, b)),A^2 -> H(U1(k, a), U1(k, b))) (A^2 -> H(U1(c, c), U1(k, b)),A^2 -> H(U1(c, c), U1(k, b))) (A^2 -> H(U1(c, d), U1(k, b)),A^2 -> H(U1(c, d), U1(k, b))) (A^2 -> H(U1(c, a), U1(k, c)),A^2 -> H(U1(c, a), U1(k, c))) (A^2 -> H(U1(c, a), U1(k, d)),A^2 -> H(U1(c, a), U1(k, d))) ---------------------------------------- (606) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(c, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (607) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(c, d)),A^2 -> H(U1(e, a), U1(c, d))) (A^2 -> H(U1(k, a), U1(c, d)),A^2 -> H(U1(k, a), U1(c, d))) (A^2 -> H(U1(c, c), U1(c, d)),A^2 -> H(U1(c, c), U1(c, d))) (A^2 -> H(U1(c, d), U1(c, d)),A^2 -> H(U1(c, d), U1(c, d))) (A^2 -> H(U1(c, a), U1(e, d)),A^2 -> H(U1(c, a), U1(e, d))) (A^2 -> H(U1(c, a), U1(k, d)),A^2 -> H(U1(c, a), U1(k, d))) (A^2 -> H(U1(c, a), U1(c, k)),A^2 -> H(U1(c, a), U1(c, k))) ---------------------------------------- (608) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(d, c)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (609) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(d, c)),A^2 -> H(U1(e, a), U1(d, c))) (A^2 -> H(U1(k, a), U1(d, c)),A^2 -> H(U1(k, a), U1(d, c))) (A^2 -> H(U1(c, c), U1(d, c)),A^2 -> H(U1(c, c), U1(d, c))) (A^2 -> H(U1(c, d), U1(d, c)),A^2 -> H(U1(c, d), U1(d, c))) (A^2 -> H(U1(c, a), U1(k, c)),A^2 -> H(U1(c, a), U1(k, c))) (A^2 -> H(U1(c, a), U1(d, e)),A^2 -> H(U1(c, a), U1(d, e))) (A^2 -> H(U1(c, a), U1(d, k)),A^2 -> H(U1(c, a), U1(d, k))) ---------------------------------------- (610) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(b, e)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (611) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(b, e)),A^2 -> H(U1(e, a), U1(b, e))) (A^2 -> H(U1(k, a), U1(b, e)),A^2 -> H(U1(k, a), U1(b, e))) (A^2 -> H(U1(c, c), U1(b, e)),A^2 -> H(U1(c, c), U1(b, e))) (A^2 -> H(U1(c, d), U1(b, e)),A^2 -> H(U1(c, d), U1(b, e))) (A^2 -> H(U1(c, a), U1(c, e)),A^2 -> H(U1(c, a), U1(c, e))) (A^2 -> H(U1(c, a), U1(d, e)),A^2 -> H(U1(c, a), U1(d, e))) ---------------------------------------- (612) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(b, k)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (613) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(b, k)),A^2 -> H(U1(e, a), U1(b, k))) (A^2 -> H(U1(k, a), U1(b, k)),A^2 -> H(U1(k, a), U1(b, k))) (A^2 -> H(U1(c, c), U1(b, k)),A^2 -> H(U1(c, c), U1(b, k))) (A^2 -> H(U1(c, d), U1(b, k)),A^2 -> H(U1(c, d), U1(b, k))) (A^2 -> H(U1(c, a), U1(c, k)),A^2 -> H(U1(c, a), U1(c, k))) (A^2 -> H(U1(c, a), U1(d, k)),A^2 -> H(U1(c, a), U1(d, k))) ---------------------------------------- (614) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, c)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (615) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(e, c)),A^2 -> H(U1(e, a), U1(e, c))) (A^2 -> H(U1(k, a), U1(e, c)),A^2 -> H(U1(k, a), U1(e, c))) (A^2 -> H(U1(c, c), U1(e, c)),A^2 -> H(U1(c, c), U1(e, c))) (A^2 -> H(U1(c, d), U1(e, c)),A^2 -> H(U1(c, d), U1(e, c))) (A^2 -> H(U1(c, a), c),A^2 -> H(U1(c, a), c)) (A^2 -> H(U1(c, a), U1(e, e)),A^2 -> H(U1(c, a), U1(e, e))) (A^2 -> H(U1(c, a), U1(e, k)),A^2 -> H(U1(c, a), U1(e, k))) ---------------------------------------- (616) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(k, c)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (617) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(k, c)),A^2 -> H(U1(e, a), U1(k, c))) (A^2 -> H(U1(k, a), U1(k, c)),A^2 -> H(U1(k, a), U1(k, c))) (A^2 -> H(U1(c, c), U1(k, c)),A^2 -> H(U1(c, c), U1(k, c))) (A^2 -> H(U1(c, d), U1(k, c)),A^2 -> H(U1(c, d), U1(k, c))) (A^2 -> H(U1(c, a), U1(k, e)),A^2 -> H(U1(c, a), U1(k, e))) (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) ---------------------------------------- (618) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(c, e)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (619) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(c, e)),A^2 -> H(U1(e, a), U1(c, e))) (A^2 -> H(U1(k, a), U1(c, e)),A^2 -> H(U1(k, a), U1(c, e))) (A^2 -> H(U1(c, c), U1(c, e)),A^2 -> H(U1(c, c), U1(c, e))) (A^2 -> H(U1(c, d), U1(c, e)),A^2 -> H(U1(c, d), U1(c, e))) (A^2 -> H(U1(c, a), U1(e, e)),A^2 -> H(U1(c, a), U1(e, e))) (A^2 -> H(U1(c, a), U1(k, e)),A^2 -> H(U1(c, a), U1(k, e))) ---------------------------------------- (620) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(c, k)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (621) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(c, k)),A^2 -> H(U1(e, a), U1(c, k))) (A^2 -> H(U1(k, a), U1(c, k)),A^2 -> H(U1(k, a), U1(c, k))) (A^2 -> H(U1(c, c), U1(c, k)),A^2 -> H(U1(c, c), U1(c, k))) (A^2 -> H(U1(c, d), U1(c, k)),A^2 -> H(U1(c, d), U1(c, k))) (A^2 -> H(U1(c, a), U1(e, k)),A^2 -> H(U1(c, a), U1(e, k))) (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) ---------------------------------------- (622) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(e, e)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (623) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(e, e)),A^2 -> H(U1(e, a), U1(e, e))) (A^2 -> H(U1(k, a), U1(e, e)),A^2 -> H(U1(k, a), U1(e, e))) (A^2 -> H(U1(c, c), U1(e, e)),A^2 -> H(U1(c, c), U1(e, e))) (A^2 -> H(U1(c, d), U1(e, e)),A^2 -> H(U1(c, d), U1(e, e))) (A^2 -> H(U1(c, a), e),A^2 -> H(U1(c, a), e)) ---------------------------------------- (624) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (625) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (626) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(k, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (627) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(k, d)),A^2 -> H(U1(e, a), U1(k, d))) (A^2 -> H(U1(k, a), U1(k, d)),A^2 -> H(U1(k, a), U1(k, d))) (A^2 -> H(U1(c, c), U1(k, d)),A^2 -> H(U1(c, c), U1(k, d))) (A^2 -> H(U1(c, d), U1(k, d)),A^2 -> H(U1(c, d), U1(k, d))) (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) ---------------------------------------- (628) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, a), U1(d, k)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (629) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(d, k)),A^2 -> H(U1(e, a), U1(d, k))) (A^2 -> H(U1(k, a), U1(d, k)),A^2 -> H(U1(k, a), U1(d, k))) (A^2 -> H(U1(c, c), U1(d, k)),A^2 -> H(U1(c, c), U1(d, k))) (A^2 -> H(U1(c, d), U1(d, k)),A^2 -> H(U1(c, d), U1(d, k))) (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) ---------------------------------------- (630) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, e), U1(b, b)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (631) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, b)),A^2 -> H(U1(k, e), U1(b, b))) (A^2 -> H(U1(d, e), U1(c, b)),A^2 -> H(U1(d, e), U1(c, b))) (A^2 -> H(U1(d, e), U1(d, b)),A^2 -> H(U1(d, e), U1(d, b))) (A^2 -> H(U1(d, e), U1(b, c)),A^2 -> H(U1(d, e), U1(b, c))) (A^2 -> H(U1(d, e), U1(b, d)),A^2 -> H(U1(d, e), U1(b, d))) ---------------------------------------- (632) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, e), f(c)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (633) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(c)),A^2 -> H(U1(k, e), f(c))) (A^2 -> H(U1(d, e), U1(c, c)),A^2 -> H(U1(d, e), U1(c, c))) (A^2 -> H(U1(d, e), f(e)),A^2 -> H(U1(d, e), f(e))) (A^2 -> H(U1(d, e), f(k)),A^2 -> H(U1(d, e), f(k))) ---------------------------------------- (634) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, e), f(d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (635) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(d)),A^2 -> H(U1(k, e), f(d))) (A^2 -> H(U1(d, e), U1(d, d)),A^2 -> H(U1(d, e), U1(d, d))) (A^2 -> H(U1(d, e), f(k)),A^2 -> H(U1(d, e), f(k))) ---------------------------------------- (636) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), U1(c, b)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (637) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, b)),A^2 -> H(U1(k, c), U1(c, b))) (A^2 -> H(U1(d, e), U1(c, b)),A^2 -> H(U1(d, e), U1(c, b))) (A^2 -> H(U1(d, k), U1(c, b)),A^2 -> H(U1(d, k), U1(c, b))) (A^2 -> H(U1(d, c), U1(e, b)),A^2 -> H(U1(d, c), U1(e, b))) (A^2 -> H(U1(d, c), U1(k, b)),A^2 -> H(U1(d, c), U1(k, b))) (A^2 -> H(U1(d, c), U1(c, c)),A^2 -> H(U1(d, c), U1(c, c))) (A^2 -> H(U1(d, c), U1(c, d)),A^2 -> H(U1(d, c), U1(c, d))) ---------------------------------------- (638) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), U1(d, b)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (639) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, b)),A^2 -> H(U1(k, c), U1(d, b))) (A^2 -> H(U1(d, e), U1(d, b)),A^2 -> H(U1(d, e), U1(d, b))) (A^2 -> H(U1(d, k), U1(d, b)),A^2 -> H(U1(d, k), U1(d, b))) (A^2 -> H(U1(d, c), U1(k, b)),A^2 -> H(U1(d, c), U1(k, b))) (A^2 -> H(U1(d, c), U1(d, c)),A^2 -> H(U1(d, c), U1(d, c))) (A^2 -> H(U1(d, c), U1(d, d)),A^2 -> H(U1(d, c), U1(d, d))) ---------------------------------------- (640) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), U1(b, c)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (641) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, c)),A^2 -> H(U1(k, c), U1(b, c))) (A^2 -> H(U1(d, e), U1(b, c)),A^2 -> H(U1(d, e), U1(b, c))) (A^2 -> H(U1(d, k), U1(b, c)),A^2 -> H(U1(d, k), U1(b, c))) (A^2 -> H(U1(d, c), U1(c, c)),A^2 -> H(U1(d, c), U1(c, c))) (A^2 -> H(U1(d, c), U1(d, c)),A^2 -> H(U1(d, c), U1(d, c))) (A^2 -> H(U1(d, c), U1(b, e)),A^2 -> H(U1(d, c), U1(b, e))) (A^2 -> H(U1(d, c), U1(b, k)),A^2 -> H(U1(d, c), U1(b, k))) ---------------------------------------- (642) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), U1(b, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (643) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, d)),A^2 -> H(U1(k, c), U1(b, d))) (A^2 -> H(U1(d, e), U1(b, d)),A^2 -> H(U1(d, e), U1(b, d))) (A^2 -> H(U1(d, k), U1(b, d)),A^2 -> H(U1(d, k), U1(b, d))) (A^2 -> H(U1(d, c), U1(c, d)),A^2 -> H(U1(d, c), U1(c, d))) (A^2 -> H(U1(d, c), U1(d, d)),A^2 -> H(U1(d, c), U1(d, d))) (A^2 -> H(U1(d, c), U1(b, k)),A^2 -> H(U1(d, c), U1(b, k))) ---------------------------------------- (644) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), U1(c, c)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (645) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, c)),A^2 -> H(U1(k, c), U1(c, c))) (A^2 -> H(U1(d, e), U1(c, c)),A^2 -> H(U1(d, e), U1(c, c))) (A^2 -> H(U1(d, k), U1(c, c)),A^2 -> H(U1(d, k), U1(c, c))) (A^2 -> H(U1(d, c), U1(e, c)),A^2 -> H(U1(d, c), U1(e, c))) (A^2 -> H(U1(d, c), U1(k, c)),A^2 -> H(U1(d, c), U1(k, c))) (A^2 -> H(U1(d, c), U1(c, e)),A^2 -> H(U1(d, c), U1(c, e))) (A^2 -> H(U1(d, c), U1(c, k)),A^2 -> H(U1(d, c), U1(c, k))) ---------------------------------------- (646) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), f(e)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (647) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(e)),A^2 -> H(U1(k, c), f(e))) (A^2 -> H(U1(d, e), f(e)),A^2 -> H(U1(d, e), f(e))) (A^2 -> H(U1(d, k), f(e)),A^2 -> H(U1(d, k), f(e))) (A^2 -> H(U1(d, c), U1(e, e)),A^2 -> H(U1(d, c), U1(e, e))) ---------------------------------------- (648) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), f(k)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (649) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), f(k)),A^2 -> H(U1(k, c), f(k))) (A^2 -> H(U1(d, e), f(k)),A^2 -> H(U1(d, e), f(k))) (A^2 -> H(U1(d, k), f(k)),A^2 -> H(U1(d, k), f(k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) ---------------------------------------- (650) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, c), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (651) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, d)),A^2 -> H(U1(k, c), U1(d, d))) (A^2 -> H(U1(d, e), U1(d, d)),A^2 -> H(U1(d, e), U1(d, d))) (A^2 -> H(U1(d, k), U1(d, d)),A^2 -> H(U1(d, k), U1(d, d))) (A^2 -> H(U1(d, c), U1(k, d)),A^2 -> H(U1(d, c), U1(k, d))) (A^2 -> H(U1(d, c), U1(d, k)),A^2 -> H(U1(d, c), U1(d, k))) ---------------------------------------- (652) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(e, b)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (653) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(e, b)),A^2 -> H(U1(k, a), U1(e, b))) (A^2 -> H(U1(d, c), U1(e, b)),A^2 -> H(U1(d, c), U1(e, b))) (A^2 -> H(U1(d, d), U1(e, b)),A^2 -> H(U1(d, d), U1(e, b))) (A^2 -> H(U1(d, a), b),A^2 -> H(U1(d, a), b)) (A^2 -> H(U1(d, a), U1(e, c)),A^2 -> H(U1(d, a), U1(e, c))) (A^2 -> H(U1(d, a), U1(e, d)),A^2 -> H(U1(d, a), U1(e, d))) ---------------------------------------- (654) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(k, b)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (655) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(k, b)),A^2 -> H(U1(k, a), U1(k, b))) (A^2 -> H(U1(d, c), U1(k, b)),A^2 -> H(U1(d, c), U1(k, b))) (A^2 -> H(U1(d, d), U1(k, b)),A^2 -> H(U1(d, d), U1(k, b))) (A^2 -> H(U1(d, a), U1(k, c)),A^2 -> H(U1(d, a), U1(k, c))) (A^2 -> H(U1(d, a), U1(k, d)),A^2 -> H(U1(d, a), U1(k, d))) ---------------------------------------- (656) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(c, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (657) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(c, d)),A^2 -> H(U1(k, a), U1(c, d))) (A^2 -> H(U1(d, c), U1(c, d)),A^2 -> H(U1(d, c), U1(c, d))) (A^2 -> H(U1(d, d), U1(c, d)),A^2 -> H(U1(d, d), U1(c, d))) (A^2 -> H(U1(d, a), U1(e, d)),A^2 -> H(U1(d, a), U1(e, d))) (A^2 -> H(U1(d, a), U1(k, d)),A^2 -> H(U1(d, a), U1(k, d))) (A^2 -> H(U1(d, a), U1(c, k)),A^2 -> H(U1(d, a), U1(c, k))) ---------------------------------------- (658) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(d, c)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (659) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(d, c)),A^2 -> H(U1(k, a), U1(d, c))) (A^2 -> H(U1(d, c), U1(d, c)),A^2 -> H(U1(d, c), U1(d, c))) (A^2 -> H(U1(d, d), U1(d, c)),A^2 -> H(U1(d, d), U1(d, c))) (A^2 -> H(U1(d, a), U1(k, c)),A^2 -> H(U1(d, a), U1(k, c))) (A^2 -> H(U1(d, a), U1(d, e)),A^2 -> H(U1(d, a), U1(d, e))) (A^2 -> H(U1(d, a), U1(d, k)),A^2 -> H(U1(d, a), U1(d, k))) ---------------------------------------- (660) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(b, e)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (661) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(b, e)),A^2 -> H(U1(k, a), U1(b, e))) (A^2 -> H(U1(d, c), U1(b, e)),A^2 -> H(U1(d, c), U1(b, e))) (A^2 -> H(U1(d, d), U1(b, e)),A^2 -> H(U1(d, d), U1(b, e))) (A^2 -> H(U1(d, a), U1(c, e)),A^2 -> H(U1(d, a), U1(c, e))) (A^2 -> H(U1(d, a), U1(d, e)),A^2 -> H(U1(d, a), U1(d, e))) ---------------------------------------- (662) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(b, k)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (663) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(b, k)),A^2 -> H(U1(k, a), U1(b, k))) (A^2 -> H(U1(d, c), U1(b, k)),A^2 -> H(U1(d, c), U1(b, k))) (A^2 -> H(U1(d, d), U1(b, k)),A^2 -> H(U1(d, d), U1(b, k))) (A^2 -> H(U1(d, a), U1(c, k)),A^2 -> H(U1(d, a), U1(c, k))) (A^2 -> H(U1(d, a), U1(d, k)),A^2 -> H(U1(d, a), U1(d, k))) ---------------------------------------- (664) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(e, c)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (665) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(e, c)),A^2 -> H(U1(k, a), U1(e, c))) (A^2 -> H(U1(d, c), U1(e, c)),A^2 -> H(U1(d, c), U1(e, c))) (A^2 -> H(U1(d, d), U1(e, c)),A^2 -> H(U1(d, d), U1(e, c))) (A^2 -> H(U1(d, a), c),A^2 -> H(U1(d, a), c)) (A^2 -> H(U1(d, a), U1(e, e)),A^2 -> H(U1(d, a), U1(e, e))) (A^2 -> H(U1(d, a), U1(e, k)),A^2 -> H(U1(d, a), U1(e, k))) ---------------------------------------- (666) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(k, c)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (667) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(k, c)),A^2 -> H(U1(k, a), U1(k, c))) (A^2 -> H(U1(d, c), U1(k, c)),A^2 -> H(U1(d, c), U1(k, c))) (A^2 -> H(U1(d, d), U1(k, c)),A^2 -> H(U1(d, d), U1(k, c))) (A^2 -> H(U1(d, a), U1(k, e)),A^2 -> H(U1(d, a), U1(k, e))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) ---------------------------------------- (668) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(c, e)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (669) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(c, e)),A^2 -> H(U1(k, a), U1(c, e))) (A^2 -> H(U1(d, c), U1(c, e)),A^2 -> H(U1(d, c), U1(c, e))) (A^2 -> H(U1(d, d), U1(c, e)),A^2 -> H(U1(d, d), U1(c, e))) (A^2 -> H(U1(d, a), U1(e, e)),A^2 -> H(U1(d, a), U1(e, e))) (A^2 -> H(U1(d, a), U1(k, e)),A^2 -> H(U1(d, a), U1(k, e))) ---------------------------------------- (670) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(c, k)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (671) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(c, k)),A^2 -> H(U1(k, a), U1(c, k))) (A^2 -> H(U1(d, c), U1(c, k)),A^2 -> H(U1(d, c), U1(c, k))) (A^2 -> H(U1(d, d), U1(c, k)),A^2 -> H(U1(d, d), U1(c, k))) (A^2 -> H(U1(d, a), U1(e, k)),A^2 -> H(U1(d, a), U1(e, k))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) ---------------------------------------- (672) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(e, e)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (673) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(e, e)),A^2 -> H(U1(k, a), U1(e, e))) (A^2 -> H(U1(d, c), U1(e, e)),A^2 -> H(U1(d, c), U1(e, e))) (A^2 -> H(U1(d, d), U1(e, e)),A^2 -> H(U1(d, d), U1(e, e))) (A^2 -> H(U1(d, a), e),A^2 -> H(U1(d, a), e)) ---------------------------------------- (674) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (675) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (676) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(k, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (677) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(k, d)),A^2 -> H(U1(k, a), U1(k, d))) (A^2 -> H(U1(d, c), U1(k, d)),A^2 -> H(U1(d, c), U1(k, d))) (A^2 -> H(U1(d, d), U1(k, d)),A^2 -> H(U1(d, d), U1(k, d))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) ---------------------------------------- (678) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, a), U1(d, k)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (679) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(d, k)),A^2 -> H(U1(k, a), U1(d, k))) (A^2 -> H(U1(d, c), U1(d, k)),A^2 -> H(U1(d, c), U1(d, k))) (A^2 -> H(U1(d, d), U1(d, k)),A^2 -> H(U1(d, d), U1(d, k))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) ---------------------------------------- (680) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), U1(c, b)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (681) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(c, b)),A^2 -> H(U1(c, e), U1(c, b))) (A^2 -> H(U1(d, e), U1(c, b)),A^2 -> H(U1(d, e), U1(c, b))) (A^2 -> H(U1(a, e), U1(e, b)),A^2 -> H(U1(a, e), U1(e, b))) (A^2 -> H(U1(a, e), U1(k, b)),A^2 -> H(U1(a, e), U1(k, b))) (A^2 -> H(U1(a, e), U1(c, c)),A^2 -> H(U1(a, e), U1(c, c))) (A^2 -> H(U1(a, e), U1(c, d)),A^2 -> H(U1(a, e), U1(c, d))) ---------------------------------------- (682) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), U1(d, b)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (683) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(d, b)),A^2 -> H(U1(c, e), U1(d, b))) (A^2 -> H(U1(d, e), U1(d, b)),A^2 -> H(U1(d, e), U1(d, b))) (A^2 -> H(U1(a, e), U1(k, b)),A^2 -> H(U1(a, e), U1(k, b))) (A^2 -> H(U1(a, e), U1(d, c)),A^2 -> H(U1(a, e), U1(d, c))) (A^2 -> H(U1(a, e), U1(d, d)),A^2 -> H(U1(a, e), U1(d, d))) ---------------------------------------- (684) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), U1(b, c)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (685) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(b, c)),A^2 -> H(U1(c, e), U1(b, c))) (A^2 -> H(U1(d, e), U1(b, c)),A^2 -> H(U1(d, e), U1(b, c))) (A^2 -> H(U1(a, e), U1(c, c)),A^2 -> H(U1(a, e), U1(c, c))) (A^2 -> H(U1(a, e), U1(d, c)),A^2 -> H(U1(a, e), U1(d, c))) (A^2 -> H(U1(a, e), U1(b, e)),A^2 -> H(U1(a, e), U1(b, e))) (A^2 -> H(U1(a, e), U1(b, k)),A^2 -> H(U1(a, e), U1(b, k))) ---------------------------------------- (686) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), U1(b, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (687) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(b, d)),A^2 -> H(U1(c, e), U1(b, d))) (A^2 -> H(U1(d, e), U1(b, d)),A^2 -> H(U1(d, e), U1(b, d))) (A^2 -> H(U1(a, e), U1(c, d)),A^2 -> H(U1(a, e), U1(c, d))) (A^2 -> H(U1(a, e), U1(d, d)),A^2 -> H(U1(a, e), U1(d, d))) (A^2 -> H(U1(a, e), U1(b, k)),A^2 -> H(U1(a, e), U1(b, k))) ---------------------------------------- (688) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), U1(c, c)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (689) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(c, c)),A^2 -> H(U1(c, e), U1(c, c))) (A^2 -> H(U1(d, e), U1(c, c)),A^2 -> H(U1(d, e), U1(c, c))) (A^2 -> H(U1(a, e), U1(e, c)),A^2 -> H(U1(a, e), U1(e, c))) (A^2 -> H(U1(a, e), U1(k, c)),A^2 -> H(U1(a, e), U1(k, c))) (A^2 -> H(U1(a, e), U1(c, e)),A^2 -> H(U1(a, e), U1(c, e))) (A^2 -> H(U1(a, e), U1(c, k)),A^2 -> H(U1(a, e), U1(c, k))) ---------------------------------------- (690) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), f(e)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (691) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), f(e)),A^2 -> H(U1(c, e), f(e))) (A^2 -> H(U1(d, e), f(e)),A^2 -> H(U1(d, e), f(e))) (A^2 -> H(U1(a, e), U1(e, e)),A^2 -> H(U1(a, e), U1(e, e))) ---------------------------------------- (692) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), f(k)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (693) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), f(k)),A^2 -> H(U1(c, e), f(k))) (A^2 -> H(U1(d, e), f(k)),A^2 -> H(U1(d, e), f(k))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) ---------------------------------------- (694) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, e), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (695) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(d, d)),A^2 -> H(U1(c, e), U1(d, d))) (A^2 -> H(U1(d, e), U1(d, d)),A^2 -> H(U1(d, e), U1(d, d))) (A^2 -> H(U1(a, e), U1(k, d)),A^2 -> H(U1(a, e), U1(k, d))) (A^2 -> H(U1(a, e), U1(d, k)),A^2 -> H(U1(a, e), U1(d, k))) ---------------------------------------- (696) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), U1(c, b)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (697) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(c, b)),A^2 -> H(U1(c, k), U1(c, b))) (A^2 -> H(U1(d, k), U1(c, b)),A^2 -> H(U1(d, k), U1(c, b))) (A^2 -> H(U1(a, k), U1(e, b)),A^2 -> H(U1(a, k), U1(e, b))) (A^2 -> H(U1(a, k), U1(k, b)),A^2 -> H(U1(a, k), U1(k, b))) (A^2 -> H(U1(a, k), U1(c, c)),A^2 -> H(U1(a, k), U1(c, c))) (A^2 -> H(U1(a, k), U1(c, d)),A^2 -> H(U1(a, k), U1(c, d))) ---------------------------------------- (698) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), U1(d, b)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (699) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(d, b)),A^2 -> H(U1(c, k), U1(d, b))) (A^2 -> H(U1(d, k), U1(d, b)),A^2 -> H(U1(d, k), U1(d, b))) (A^2 -> H(U1(a, k), U1(k, b)),A^2 -> H(U1(a, k), U1(k, b))) (A^2 -> H(U1(a, k), U1(d, c)),A^2 -> H(U1(a, k), U1(d, c))) (A^2 -> H(U1(a, k), U1(d, d)),A^2 -> H(U1(a, k), U1(d, d))) ---------------------------------------- (700) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), U1(b, c)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (701) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(b, c)),A^2 -> H(U1(c, k), U1(b, c))) (A^2 -> H(U1(d, k), U1(b, c)),A^2 -> H(U1(d, k), U1(b, c))) (A^2 -> H(U1(a, k), U1(c, c)),A^2 -> H(U1(a, k), U1(c, c))) (A^2 -> H(U1(a, k), U1(d, c)),A^2 -> H(U1(a, k), U1(d, c))) (A^2 -> H(U1(a, k), U1(b, e)),A^2 -> H(U1(a, k), U1(b, e))) (A^2 -> H(U1(a, k), U1(b, k)),A^2 -> H(U1(a, k), U1(b, k))) ---------------------------------------- (702) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), U1(b, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (703) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(b, d)),A^2 -> H(U1(c, k), U1(b, d))) (A^2 -> H(U1(d, k), U1(b, d)),A^2 -> H(U1(d, k), U1(b, d))) (A^2 -> H(U1(a, k), U1(c, d)),A^2 -> H(U1(a, k), U1(c, d))) (A^2 -> H(U1(a, k), U1(d, d)),A^2 -> H(U1(a, k), U1(d, d))) (A^2 -> H(U1(a, k), U1(b, k)),A^2 -> H(U1(a, k), U1(b, k))) ---------------------------------------- (704) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), U1(c, c)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (705) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(c, c)),A^2 -> H(U1(c, k), U1(c, c))) (A^2 -> H(U1(d, k), U1(c, c)),A^2 -> H(U1(d, k), U1(c, c))) (A^2 -> H(U1(a, k), U1(e, c)),A^2 -> H(U1(a, k), U1(e, c))) (A^2 -> H(U1(a, k), U1(k, c)),A^2 -> H(U1(a, k), U1(k, c))) (A^2 -> H(U1(a, k), U1(c, e)),A^2 -> H(U1(a, k), U1(c, e))) (A^2 -> H(U1(a, k), U1(c, k)),A^2 -> H(U1(a, k), U1(c, k))) ---------------------------------------- (706) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), f(e)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (707) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), f(e)),A^2 -> H(U1(c, k), f(e))) (A^2 -> H(U1(d, k), f(e)),A^2 -> H(U1(d, k), f(e))) (A^2 -> H(U1(a, k), U1(e, e)),A^2 -> H(U1(a, k), U1(e, e))) ---------------------------------------- (708) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), f(k)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (709) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), f(k)),A^2 -> H(U1(c, k), f(k))) (A^2 -> H(U1(d, k), f(k)),A^2 -> H(U1(d, k), f(k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (710) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, k), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (711) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(d, d)),A^2 -> H(U1(c, k), U1(d, d))) (A^2 -> H(U1(d, k), U1(d, d)),A^2 -> H(U1(d, k), U1(d, d))) (A^2 -> H(U1(a, k), U1(k, d)),A^2 -> H(U1(a, k), U1(k, d))) (A^2 -> H(U1(a, k), U1(d, k)),A^2 -> H(U1(a, k), U1(d, k))) ---------------------------------------- (712) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(e, b)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (713) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, b)),A^2 -> H(U1(c, c), U1(e, b))) (A^2 -> H(U1(d, c), U1(e, b)),A^2 -> H(U1(d, c), U1(e, b))) (A^2 -> H(U1(a, e), U1(e, b)),A^2 -> H(U1(a, e), U1(e, b))) (A^2 -> H(U1(a, k), U1(e, b)),A^2 -> H(U1(a, k), U1(e, b))) (A^2 -> H(U1(a, c), b),A^2 -> H(U1(a, c), b)) (A^2 -> H(U1(a, c), U1(e, c)),A^2 -> H(U1(a, c), U1(e, c))) (A^2 -> H(U1(a, c), U1(e, d)),A^2 -> H(U1(a, c), U1(e, d))) ---------------------------------------- (714) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(k, b)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (715) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, b)),A^2 -> H(U1(c, c), U1(k, b))) (A^2 -> H(U1(d, c), U1(k, b)),A^2 -> H(U1(d, c), U1(k, b))) (A^2 -> H(U1(a, e), U1(k, b)),A^2 -> H(U1(a, e), U1(k, b))) (A^2 -> H(U1(a, k), U1(k, b)),A^2 -> H(U1(a, k), U1(k, b))) (A^2 -> H(U1(a, c), U1(k, c)),A^2 -> H(U1(a, c), U1(k, c))) (A^2 -> H(U1(a, c), U1(k, d)),A^2 -> H(U1(a, c), U1(k, d))) ---------------------------------------- (716) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(c, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (717) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, d)),A^2 -> H(U1(c, c), U1(c, d))) (A^2 -> H(U1(d, c), U1(c, d)),A^2 -> H(U1(d, c), U1(c, d))) (A^2 -> H(U1(a, e), U1(c, d)),A^2 -> H(U1(a, e), U1(c, d))) (A^2 -> H(U1(a, k), U1(c, d)),A^2 -> H(U1(a, k), U1(c, d))) (A^2 -> H(U1(a, c), U1(e, d)),A^2 -> H(U1(a, c), U1(e, d))) (A^2 -> H(U1(a, c), U1(k, d)),A^2 -> H(U1(a, c), U1(k, d))) (A^2 -> H(U1(a, c), U1(c, k)),A^2 -> H(U1(a, c), U1(c, k))) ---------------------------------------- (718) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(d, c)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (719) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, c)),A^2 -> H(U1(c, c), U1(d, c))) (A^2 -> H(U1(d, c), U1(d, c)),A^2 -> H(U1(d, c), U1(d, c))) (A^2 -> H(U1(a, e), U1(d, c)),A^2 -> H(U1(a, e), U1(d, c))) (A^2 -> H(U1(a, k), U1(d, c)),A^2 -> H(U1(a, k), U1(d, c))) (A^2 -> H(U1(a, c), U1(k, c)),A^2 -> H(U1(a, c), U1(k, c))) (A^2 -> H(U1(a, c), U1(d, e)),A^2 -> H(U1(a, c), U1(d, e))) (A^2 -> H(U1(a, c), U1(d, k)),A^2 -> H(U1(a, c), U1(d, k))) ---------------------------------------- (720) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(b, e)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (721) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, e)),A^2 -> H(U1(c, c), U1(b, e))) (A^2 -> H(U1(d, c), U1(b, e)),A^2 -> H(U1(d, c), U1(b, e))) (A^2 -> H(U1(a, e), U1(b, e)),A^2 -> H(U1(a, e), U1(b, e))) (A^2 -> H(U1(a, k), U1(b, e)),A^2 -> H(U1(a, k), U1(b, e))) (A^2 -> H(U1(a, c), U1(c, e)),A^2 -> H(U1(a, c), U1(c, e))) (A^2 -> H(U1(a, c), U1(d, e)),A^2 -> H(U1(a, c), U1(d, e))) ---------------------------------------- (722) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(b, k)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (723) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(b, k)),A^2 -> H(U1(c, c), U1(b, k))) (A^2 -> H(U1(d, c), U1(b, k)),A^2 -> H(U1(d, c), U1(b, k))) (A^2 -> H(U1(a, e), U1(b, k)),A^2 -> H(U1(a, e), U1(b, k))) (A^2 -> H(U1(a, k), U1(b, k)),A^2 -> H(U1(a, k), U1(b, k))) (A^2 -> H(U1(a, c), U1(c, k)),A^2 -> H(U1(a, c), U1(c, k))) (A^2 -> H(U1(a, c), U1(d, k)),A^2 -> H(U1(a, c), U1(d, k))) ---------------------------------------- (724) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(e, c)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (725) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, c)),A^2 -> H(U1(c, c), U1(e, c))) (A^2 -> H(U1(d, c), U1(e, c)),A^2 -> H(U1(d, c), U1(e, c))) (A^2 -> H(U1(a, e), U1(e, c)),A^2 -> H(U1(a, e), U1(e, c))) (A^2 -> H(U1(a, k), U1(e, c)),A^2 -> H(U1(a, k), U1(e, c))) (A^2 -> H(U1(a, c), c),A^2 -> H(U1(a, c), c)) (A^2 -> H(U1(a, c), U1(e, e)),A^2 -> H(U1(a, c), U1(e, e))) (A^2 -> H(U1(a, c), U1(e, k)),A^2 -> H(U1(a, c), U1(e, k))) ---------------------------------------- (726) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(k, c)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (727) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, c)),A^2 -> H(U1(c, c), U1(k, c))) (A^2 -> H(U1(d, c), U1(k, c)),A^2 -> H(U1(d, c), U1(k, c))) (A^2 -> H(U1(a, e), U1(k, c)),A^2 -> H(U1(a, e), U1(k, c))) (A^2 -> H(U1(a, k), U1(k, c)),A^2 -> H(U1(a, k), U1(k, c))) (A^2 -> H(U1(a, c), U1(k, e)),A^2 -> H(U1(a, c), U1(k, e))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) ---------------------------------------- (728) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(c, e)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (729) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, e)),A^2 -> H(U1(c, c), U1(c, e))) (A^2 -> H(U1(d, c), U1(c, e)),A^2 -> H(U1(d, c), U1(c, e))) (A^2 -> H(U1(a, e), U1(c, e)),A^2 -> H(U1(a, e), U1(c, e))) (A^2 -> H(U1(a, k), U1(c, e)),A^2 -> H(U1(a, k), U1(c, e))) (A^2 -> H(U1(a, c), U1(e, e)),A^2 -> H(U1(a, c), U1(e, e))) (A^2 -> H(U1(a, c), U1(k, e)),A^2 -> H(U1(a, c), U1(k, e))) ---------------------------------------- (730) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(c, k)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (731) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(c, k)),A^2 -> H(U1(c, c), U1(c, k))) (A^2 -> H(U1(d, c), U1(c, k)),A^2 -> H(U1(d, c), U1(c, k))) (A^2 -> H(U1(a, e), U1(c, k)),A^2 -> H(U1(a, e), U1(c, k))) (A^2 -> H(U1(a, k), U1(c, k)),A^2 -> H(U1(a, k), U1(c, k))) (A^2 -> H(U1(a, c), U1(e, k)),A^2 -> H(U1(a, c), U1(e, k))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) ---------------------------------------- (732) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(e, e)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (733) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, e)),A^2 -> H(U1(c, c), U1(e, e))) (A^2 -> H(U1(d, c), U1(e, e)),A^2 -> H(U1(d, c), U1(e, e))) (A^2 -> H(U1(a, e), U1(e, e)),A^2 -> H(U1(a, e), U1(e, e))) (A^2 -> H(U1(a, k), U1(e, e)),A^2 -> H(U1(a, k), U1(e, e))) (A^2 -> H(U1(a, c), e),A^2 -> H(U1(a, c), e)) ---------------------------------------- (734) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (735) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (736) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(k, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (737) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, d)),A^2 -> H(U1(c, c), U1(k, d))) (A^2 -> H(U1(d, c), U1(k, d)),A^2 -> H(U1(d, c), U1(k, d))) (A^2 -> H(U1(a, e), U1(k, d)),A^2 -> H(U1(a, e), U1(k, d))) (A^2 -> H(U1(a, k), U1(k, d)),A^2 -> H(U1(a, k), U1(k, d))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) ---------------------------------------- (738) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, c), U1(d, k)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (739) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, k)),A^2 -> H(U1(c, c), U1(d, k))) (A^2 -> H(U1(d, c), U1(d, k)),A^2 -> H(U1(d, c), U1(d, k))) (A^2 -> H(U1(a, e), U1(d, k)),A^2 -> H(U1(a, e), U1(d, k))) (A^2 -> H(U1(a, k), U1(d, k)),A^2 -> H(U1(a, k), U1(d, k))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) ---------------------------------------- (740) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(e, b)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (741) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(e, b)),A^2 -> H(U1(c, d), U1(e, b))) (A^2 -> H(U1(d, d), U1(e, b)),A^2 -> H(U1(d, d), U1(e, b))) (A^2 -> H(U1(a, k), U1(e, b)),A^2 -> H(U1(a, k), U1(e, b))) (A^2 -> H(U1(a, d), b),A^2 -> H(U1(a, d), b)) (A^2 -> H(U1(a, d), U1(e, c)),A^2 -> H(U1(a, d), U1(e, c))) (A^2 -> H(U1(a, d), U1(e, d)),A^2 -> H(U1(a, d), U1(e, d))) ---------------------------------------- (742) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(k, b)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (743) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(k, b)),A^2 -> H(U1(c, d), U1(k, b))) (A^2 -> H(U1(d, d), U1(k, b)),A^2 -> H(U1(d, d), U1(k, b))) (A^2 -> H(U1(a, k), U1(k, b)),A^2 -> H(U1(a, k), U1(k, b))) (A^2 -> H(U1(a, d), U1(k, c)),A^2 -> H(U1(a, d), U1(k, c))) (A^2 -> H(U1(a, d), U1(k, d)),A^2 -> H(U1(a, d), U1(k, d))) ---------------------------------------- (744) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(c, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (745) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(c, d)),A^2 -> H(U1(c, d), U1(c, d))) (A^2 -> H(U1(d, d), U1(c, d)),A^2 -> H(U1(d, d), U1(c, d))) (A^2 -> H(U1(a, k), U1(c, d)),A^2 -> H(U1(a, k), U1(c, d))) (A^2 -> H(U1(a, d), U1(e, d)),A^2 -> H(U1(a, d), U1(e, d))) (A^2 -> H(U1(a, d), U1(k, d)),A^2 -> H(U1(a, d), U1(k, d))) (A^2 -> H(U1(a, d), U1(c, k)),A^2 -> H(U1(a, d), U1(c, k))) ---------------------------------------- (746) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(d, c)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (747) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(d, c)),A^2 -> H(U1(c, d), U1(d, c))) (A^2 -> H(U1(d, d), U1(d, c)),A^2 -> H(U1(d, d), U1(d, c))) (A^2 -> H(U1(a, k), U1(d, c)),A^2 -> H(U1(a, k), U1(d, c))) (A^2 -> H(U1(a, d), U1(k, c)),A^2 -> H(U1(a, d), U1(k, c))) (A^2 -> H(U1(a, d), U1(d, e)),A^2 -> H(U1(a, d), U1(d, e))) (A^2 -> H(U1(a, d), U1(d, k)),A^2 -> H(U1(a, d), U1(d, k))) ---------------------------------------- (748) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(b, e)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (749) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(b, e)),A^2 -> H(U1(c, d), U1(b, e))) (A^2 -> H(U1(d, d), U1(b, e)),A^2 -> H(U1(d, d), U1(b, e))) (A^2 -> H(U1(a, k), U1(b, e)),A^2 -> H(U1(a, k), U1(b, e))) (A^2 -> H(U1(a, d), U1(c, e)),A^2 -> H(U1(a, d), U1(c, e))) (A^2 -> H(U1(a, d), U1(d, e)),A^2 -> H(U1(a, d), U1(d, e))) ---------------------------------------- (750) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(b, k)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (751) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(b, k)),A^2 -> H(U1(c, d), U1(b, k))) (A^2 -> H(U1(d, d), U1(b, k)),A^2 -> H(U1(d, d), U1(b, k))) (A^2 -> H(U1(a, k), U1(b, k)),A^2 -> H(U1(a, k), U1(b, k))) (A^2 -> H(U1(a, d), U1(c, k)),A^2 -> H(U1(a, d), U1(c, k))) (A^2 -> H(U1(a, d), U1(d, k)),A^2 -> H(U1(a, d), U1(d, k))) ---------------------------------------- (752) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(e, c)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (753) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(e, c)),A^2 -> H(U1(c, d), U1(e, c))) (A^2 -> H(U1(d, d), U1(e, c)),A^2 -> H(U1(d, d), U1(e, c))) (A^2 -> H(U1(a, k), U1(e, c)),A^2 -> H(U1(a, k), U1(e, c))) (A^2 -> H(U1(a, d), c),A^2 -> H(U1(a, d), c)) (A^2 -> H(U1(a, d), U1(e, e)),A^2 -> H(U1(a, d), U1(e, e))) (A^2 -> H(U1(a, d), U1(e, k)),A^2 -> H(U1(a, d), U1(e, k))) ---------------------------------------- (754) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(k, c)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (755) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(k, c)),A^2 -> H(U1(c, d), U1(k, c))) (A^2 -> H(U1(d, d), U1(k, c)),A^2 -> H(U1(d, d), U1(k, c))) (A^2 -> H(U1(a, k), U1(k, c)),A^2 -> H(U1(a, k), U1(k, c))) (A^2 -> H(U1(a, d), U1(k, e)),A^2 -> H(U1(a, d), U1(k, e))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (756) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(c, e)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (757) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(c, e)),A^2 -> H(U1(c, d), U1(c, e))) (A^2 -> H(U1(d, d), U1(c, e)),A^2 -> H(U1(d, d), U1(c, e))) (A^2 -> H(U1(a, k), U1(c, e)),A^2 -> H(U1(a, k), U1(c, e))) (A^2 -> H(U1(a, d), U1(e, e)),A^2 -> H(U1(a, d), U1(e, e))) (A^2 -> H(U1(a, d), U1(k, e)),A^2 -> H(U1(a, d), U1(k, e))) ---------------------------------------- (758) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(c, k)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (759) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(c, k)),A^2 -> H(U1(c, d), U1(c, k))) (A^2 -> H(U1(d, d), U1(c, k)),A^2 -> H(U1(d, d), U1(c, k))) (A^2 -> H(U1(a, k), U1(c, k)),A^2 -> H(U1(a, k), U1(c, k))) (A^2 -> H(U1(a, d), U1(e, k)),A^2 -> H(U1(a, d), U1(e, k))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (760) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(e, e)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (761) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(e, e)),A^2 -> H(U1(c, d), U1(e, e))) (A^2 -> H(U1(d, d), U1(e, e)),A^2 -> H(U1(d, d), U1(e, e))) (A^2 -> H(U1(a, k), U1(e, e)),A^2 -> H(U1(a, k), U1(e, e))) (A^2 -> H(U1(a, d), e),A^2 -> H(U1(a, d), e)) ---------------------------------------- (762) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (763) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (764) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(k, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (765) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(k, d)),A^2 -> H(U1(c, d), U1(k, d))) (A^2 -> H(U1(d, d), U1(k, d)),A^2 -> H(U1(d, d), U1(k, d))) (A^2 -> H(U1(a, k), U1(k, d)),A^2 -> H(U1(a, k), U1(k, d))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (766) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, d), U1(d, k)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (767) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(d, k)),A^2 -> H(U1(c, d), U1(d, k))) (A^2 -> H(U1(d, d), U1(d, k)),A^2 -> H(U1(d, d), U1(d, k))) (A^2 -> H(U1(a, k), U1(d, k)),A^2 -> H(U1(a, k), U1(d, k))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (768) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), b) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (769) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), b),A^2 -> H(U1(c, a), b)) (A^2 -> H(U1(d, a), b),A^2 -> H(U1(d, a), b)) (A^2 -> H(U1(a, c), b),A^2 -> H(U1(a, c), b)) (A^2 -> H(U1(a, d), b),A^2 -> H(U1(a, d), b)) (A^2 -> H(U1(a, a), c),A^2 -> H(U1(a, a), c)) (A^2 -> H(U1(a, a), d),A^2 -> H(U1(a, a), d)) ---------------------------------------- (770) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (771) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(e, d)),A^2 -> H(U1(c, a), U1(e, d))) (A^2 -> H(U1(d, a), U1(e, d)),A^2 -> H(U1(d, a), U1(e, d))) (A^2 -> H(U1(a, c), U1(e, d)),A^2 -> H(U1(a, c), U1(e, d))) (A^2 -> H(U1(a, d), U1(e, d)),A^2 -> H(U1(a, d), U1(e, d))) (A^2 -> H(U1(a, a), d),A^2 -> H(U1(a, a), d)) (A^2 -> H(U1(a, a), U1(e, k)),A^2 -> H(U1(a, a), U1(e, k))) ---------------------------------------- (772) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), U1(d, e)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (773) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(d, e)),A^2 -> H(U1(c, a), U1(d, e))) (A^2 -> H(U1(d, a), U1(d, e)),A^2 -> H(U1(d, a), U1(d, e))) (A^2 -> H(U1(a, c), U1(d, e)),A^2 -> H(U1(a, c), U1(d, e))) (A^2 -> H(U1(a, d), U1(d, e)),A^2 -> H(U1(a, d), U1(d, e))) (A^2 -> H(U1(a, a), U1(k, e)),A^2 -> H(U1(a, a), U1(k, e))) ---------------------------------------- (774) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), c) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (775) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), c),A^2 -> H(U1(c, a), c)) (A^2 -> H(U1(d, a), c),A^2 -> H(U1(d, a), c)) (A^2 -> H(U1(a, c), c),A^2 -> H(U1(a, c), c)) (A^2 -> H(U1(a, d), c),A^2 -> H(U1(a, d), c)) (A^2 -> H(U1(a, a), e),A^2 -> H(U1(a, a), e)) (A^2 -> H(U1(a, a), k),A^2 -> H(U1(a, a), k)) ---------------------------------------- (776) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), U1(e, k)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (777) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(e, k)),A^2 -> H(U1(c, a), U1(e, k))) (A^2 -> H(U1(d, a), U1(e, k)),A^2 -> H(U1(d, a), U1(e, k))) (A^2 -> H(U1(a, c), U1(e, k)),A^2 -> H(U1(a, c), U1(e, k))) (A^2 -> H(U1(a, d), U1(e, k)),A^2 -> H(U1(a, d), U1(e, k))) (A^2 -> H(U1(a, a), k),A^2 -> H(U1(a, a), k)) ---------------------------------------- (778) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), U1(k, e)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (779) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(k, e)),A^2 -> H(U1(c, a), U1(k, e))) (A^2 -> H(U1(d, a), U1(k, e)),A^2 -> H(U1(d, a), U1(k, e))) (A^2 -> H(U1(a, c), U1(k, e)),A^2 -> H(U1(a, c), U1(k, e))) (A^2 -> H(U1(a, d), U1(k, e)),A^2 -> H(U1(a, d), U1(k, e))) ---------------------------------------- (780) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), e) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (781) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), e),A^2 -> H(U1(c, a), e)) (A^2 -> H(U1(d, a), e),A^2 -> H(U1(d, a), e)) (A^2 -> H(U1(a, c), e),A^2 -> H(U1(a, c), e)) (A^2 -> H(U1(a, d), e),A^2 -> H(U1(a, d), e)) ---------------------------------------- (782) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(a, a), U1(k, k)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (783) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), U1(k, k)),A^2 -> H(U1(c, a), U1(k, k))) (A^2 -> H(U1(d, a), U1(k, k)),A^2 -> H(U1(d, a), U1(k, k))) (A^2 -> H(U1(a, c), U1(k, k)),A^2 -> H(U1(a, c), U1(k, k))) (A^2 -> H(U1(a, d), U1(k, k)),A^2 -> H(U1(a, d), U1(k, k))) ---------------------------------------- (784) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(k, f(b)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (785) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, f(b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, b)),A^2 -> H(k, U1(b, b))) (A^2 -> H(k, f(c)),A^2 -> H(k, f(c))) (A^2 -> H(k, f(d)),A^2 -> H(k, f(d))) ---------------------------------------- (786) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(c, U1(b, b)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (787) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, b)),A^2 -> H(e, U1(b, b))) (A^2 -> H(k, U1(b, b)),A^2 -> H(k, U1(b, b))) (A^2 -> H(c, U1(c, b)),A^2 -> H(c, U1(c, b))) (A^2 -> H(c, U1(d, b)),A^2 -> H(c, U1(d, b))) (A^2 -> H(c, U1(b, c)),A^2 -> H(c, U1(b, c))) (A^2 -> H(c, U1(b, d)),A^2 -> H(c, U1(b, d))) ---------------------------------------- (788) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(c, f(c)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (789) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(c)),A^2 -> H(e, f(c))) (A^2 -> H(k, f(c)),A^2 -> H(k, f(c))) (A^2 -> H(c, U1(c, c)),A^2 -> H(c, U1(c, c))) (A^2 -> H(c, f(e)),A^2 -> H(c, f(e))) (A^2 -> H(c, f(k)),A^2 -> H(c, f(k))) ---------------------------------------- (790) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(c, f(d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (791) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(d)),A^2 -> H(e, f(d))) (A^2 -> H(k, f(d)),A^2 -> H(k, f(d))) (A^2 -> H(c, U1(d, d)),A^2 -> H(c, U1(d, d))) (A^2 -> H(c, f(k)),A^2 -> H(c, f(k))) ---------------------------------------- (792) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, k), U1(b, b)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (793) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, b)),A^2 -> H(k, U1(b, b))) (A^2 -> H(U1(e, k), U1(c, b)),A^2 -> H(U1(e, k), U1(c, b))) (A^2 -> H(U1(e, k), U1(d, b)),A^2 -> H(U1(e, k), U1(d, b))) (A^2 -> H(U1(e, k), U1(b, c)),A^2 -> H(U1(e, k), U1(b, c))) (A^2 -> H(U1(e, k), U1(b, d)),A^2 -> H(U1(e, k), U1(b, d))) ---------------------------------------- (794) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, k), f(c)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (795) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(c)),A^2 -> H(k, f(c))) (A^2 -> H(U1(e, k), U1(c, c)),A^2 -> H(U1(e, k), U1(c, c))) (A^2 -> H(U1(e, k), f(e)),A^2 -> H(U1(e, k), f(e))) (A^2 -> H(U1(e, k), f(k)),A^2 -> H(U1(e, k), f(k))) ---------------------------------------- (796) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, k), f(d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (797) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(d)),A^2 -> H(k, f(d))) (A^2 -> H(U1(e, k), U1(d, d)),A^2 -> H(U1(e, k), U1(d, d))) (A^2 -> H(U1(e, k), f(k)),A^2 -> H(U1(e, k), f(k))) ---------------------------------------- (798) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), U1(c, b)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (799) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, b)),A^2 -> H(c, U1(c, b))) (A^2 -> H(U1(e, e), U1(c, b)),A^2 -> H(U1(e, e), U1(c, b))) (A^2 -> H(U1(e, k), U1(c, b)),A^2 -> H(U1(e, k), U1(c, b))) (A^2 -> H(U1(e, c), U1(e, b)),A^2 -> H(U1(e, c), U1(e, b))) (A^2 -> H(U1(e, c), U1(k, b)),A^2 -> H(U1(e, c), U1(k, b))) (A^2 -> H(U1(e, c), U1(c, c)),A^2 -> H(U1(e, c), U1(c, c))) (A^2 -> H(U1(e, c), U1(c, d)),A^2 -> H(U1(e, c), U1(c, d))) ---------------------------------------- (800) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), U1(d, b)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (801) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, b)),A^2 -> H(c, U1(d, b))) (A^2 -> H(U1(e, e), U1(d, b)),A^2 -> H(U1(e, e), U1(d, b))) (A^2 -> H(U1(e, k), U1(d, b)),A^2 -> H(U1(e, k), U1(d, b))) (A^2 -> H(U1(e, c), U1(k, b)),A^2 -> H(U1(e, c), U1(k, b))) (A^2 -> H(U1(e, c), U1(d, c)),A^2 -> H(U1(e, c), U1(d, c))) (A^2 -> H(U1(e, c), U1(d, d)),A^2 -> H(U1(e, c), U1(d, d))) ---------------------------------------- (802) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), U1(b, c)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (803) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, c)),A^2 -> H(c, U1(b, c))) (A^2 -> H(U1(e, e), U1(b, c)),A^2 -> H(U1(e, e), U1(b, c))) (A^2 -> H(U1(e, k), U1(b, c)),A^2 -> H(U1(e, k), U1(b, c))) (A^2 -> H(U1(e, c), U1(c, c)),A^2 -> H(U1(e, c), U1(c, c))) (A^2 -> H(U1(e, c), U1(d, c)),A^2 -> H(U1(e, c), U1(d, c))) (A^2 -> H(U1(e, c), U1(b, e)),A^2 -> H(U1(e, c), U1(b, e))) (A^2 -> H(U1(e, c), U1(b, k)),A^2 -> H(U1(e, c), U1(b, k))) ---------------------------------------- (804) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), U1(b, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (805) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, d)),A^2 -> H(c, U1(b, d))) (A^2 -> H(U1(e, e), U1(b, d)),A^2 -> H(U1(e, e), U1(b, d))) (A^2 -> H(U1(e, k), U1(b, d)),A^2 -> H(U1(e, k), U1(b, d))) (A^2 -> H(U1(e, c), U1(c, d)),A^2 -> H(U1(e, c), U1(c, d))) (A^2 -> H(U1(e, c), U1(d, d)),A^2 -> H(U1(e, c), U1(d, d))) (A^2 -> H(U1(e, c), U1(b, k)),A^2 -> H(U1(e, c), U1(b, k))) ---------------------------------------- (806) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), U1(c, c)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (807) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, c)),A^2 -> H(c, U1(c, c))) (A^2 -> H(U1(e, e), U1(c, c)),A^2 -> H(U1(e, e), U1(c, c))) (A^2 -> H(U1(e, k), U1(c, c)),A^2 -> H(U1(e, k), U1(c, c))) (A^2 -> H(U1(e, c), U1(e, c)),A^2 -> H(U1(e, c), U1(e, c))) (A^2 -> H(U1(e, c), U1(k, c)),A^2 -> H(U1(e, c), U1(k, c))) (A^2 -> H(U1(e, c), U1(c, e)),A^2 -> H(U1(e, c), U1(c, e))) (A^2 -> H(U1(e, c), U1(c, k)),A^2 -> H(U1(e, c), U1(c, k))) ---------------------------------------- (808) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), f(e)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (809) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(e)),A^2 -> H(c, f(e))) (A^2 -> H(U1(e, e), f(e)),A^2 -> H(U1(e, e), f(e))) (A^2 -> H(U1(e, k), f(e)),A^2 -> H(U1(e, k), f(e))) (A^2 -> H(U1(e, c), U1(e, e)),A^2 -> H(U1(e, c), U1(e, e))) ---------------------------------------- (810) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), f(k)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (811) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(k)),A^2 -> H(c, f(k))) (A^2 -> H(U1(e, e), f(k)),A^2 -> H(U1(e, e), f(k))) (A^2 -> H(U1(e, k), f(k)),A^2 -> H(U1(e, k), f(k))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) ---------------------------------------- (812) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, c), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (813) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, d)),A^2 -> H(c, U1(d, d))) (A^2 -> H(U1(e, e), U1(d, d)),A^2 -> H(U1(e, e), U1(d, d))) (A^2 -> H(U1(e, k), U1(d, d)),A^2 -> H(U1(e, k), U1(d, d))) (A^2 -> H(U1(e, c), U1(k, d)),A^2 -> H(U1(e, c), U1(k, d))) (A^2 -> H(U1(e, c), U1(d, k)),A^2 -> H(U1(e, c), U1(d, k))) ---------------------------------------- (814) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, e), U1(b, b)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (815) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, b)),A^2 -> H(U1(k, e), U1(c, b))) (A^2 -> H(U1(k, e), U1(d, b)),A^2 -> H(U1(k, e), U1(d, b))) (A^2 -> H(U1(k, e), U1(b, c)),A^2 -> H(U1(k, e), U1(b, c))) (A^2 -> H(U1(k, e), U1(b, d)),A^2 -> H(U1(k, e), U1(b, d))) ---------------------------------------- (816) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, e), f(c)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (817) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, c)),A^2 -> H(U1(k, e), U1(c, c))) (A^2 -> H(U1(k, e), f(e)),A^2 -> H(U1(k, e), f(e))) (A^2 -> H(U1(k, e), f(k)),A^2 -> H(U1(k, e), f(k))) ---------------------------------------- (818) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, e), f(d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (819) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, d)),A^2 -> H(U1(k, e), U1(d, d))) (A^2 -> H(U1(k, e), f(k)),A^2 -> H(U1(k, e), f(k))) ---------------------------------------- (820) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), U1(c, b)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (821) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, b)),A^2 -> H(U1(k, e), U1(c, b))) (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(U1(k, c), U1(e, b)),A^2 -> H(U1(k, c), U1(e, b))) (A^2 -> H(U1(k, c), U1(k, b)),A^2 -> H(U1(k, c), U1(k, b))) (A^2 -> H(U1(k, c), U1(c, c)),A^2 -> H(U1(k, c), U1(c, c))) (A^2 -> H(U1(k, c), U1(c, d)),A^2 -> H(U1(k, c), U1(c, d))) ---------------------------------------- (822) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), U1(d, b)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (823) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, b)),A^2 -> H(U1(k, e), U1(d, b))) (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(U1(k, c), U1(k, b)),A^2 -> H(U1(k, c), U1(k, b))) (A^2 -> H(U1(k, c), U1(d, c)),A^2 -> H(U1(k, c), U1(d, c))) (A^2 -> H(U1(k, c), U1(d, d)),A^2 -> H(U1(k, c), U1(d, d))) ---------------------------------------- (824) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), U1(b, c)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (825) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, c)),A^2 -> H(U1(k, e), U1(b, c))) (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(U1(k, c), U1(c, c)),A^2 -> H(U1(k, c), U1(c, c))) (A^2 -> H(U1(k, c), U1(d, c)),A^2 -> H(U1(k, c), U1(d, c))) (A^2 -> H(U1(k, c), U1(b, e)),A^2 -> H(U1(k, c), U1(b, e))) (A^2 -> H(U1(k, c), U1(b, k)),A^2 -> H(U1(k, c), U1(b, k))) ---------------------------------------- (826) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), U1(b, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (827) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, d)),A^2 -> H(U1(k, e), U1(b, d))) (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) (A^2 -> H(U1(k, c), U1(c, d)),A^2 -> H(U1(k, c), U1(c, d))) (A^2 -> H(U1(k, c), U1(d, d)),A^2 -> H(U1(k, c), U1(d, d))) (A^2 -> H(U1(k, c), U1(b, k)),A^2 -> H(U1(k, c), U1(b, k))) ---------------------------------------- (828) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), U1(c, c)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (829) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, c)),A^2 -> H(U1(k, e), U1(c, c))) (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, c), U1(e, c)),A^2 -> H(U1(k, c), U1(e, c))) (A^2 -> H(U1(k, c), U1(k, c)),A^2 -> H(U1(k, c), U1(k, c))) (A^2 -> H(U1(k, c), U1(c, e)),A^2 -> H(U1(k, c), U1(c, e))) (A^2 -> H(U1(k, c), U1(c, k)),A^2 -> H(U1(k, c), U1(c, k))) ---------------------------------------- (830) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), f(e)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (831) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(e)),A^2 -> H(U1(k, e), f(e))) (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) ---------------------------------------- (832) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), f(k)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (833) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(k)),A^2 -> H(U1(k, e), f(k))) (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) ---------------------------------------- (834) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, c), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (835) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, d)),A^2 -> H(U1(k, e), U1(d, d))) (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(k, c), U1(k, d)),A^2 -> H(U1(k, c), U1(k, d))) (A^2 -> H(U1(k, c), U1(d, k)),A^2 -> H(U1(k, c), U1(d, k))) ---------------------------------------- (836) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), U1(c, b)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (837) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, b)),A^2 -> H(U1(e, e), U1(c, b))) (A^2 -> H(U1(k, e), U1(c, b)),A^2 -> H(U1(k, e), U1(c, b))) (A^2 -> H(U1(c, e), U1(e, b)),A^2 -> H(U1(c, e), U1(e, b))) (A^2 -> H(U1(c, e), U1(k, b)),A^2 -> H(U1(c, e), U1(k, b))) (A^2 -> H(U1(c, e), U1(c, c)),A^2 -> H(U1(c, e), U1(c, c))) (A^2 -> H(U1(c, e), U1(c, d)),A^2 -> H(U1(c, e), U1(c, d))) ---------------------------------------- (838) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), U1(d, b)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (839) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, b)),A^2 -> H(U1(e, e), U1(d, b))) (A^2 -> H(U1(k, e), U1(d, b)),A^2 -> H(U1(k, e), U1(d, b))) (A^2 -> H(U1(c, e), U1(k, b)),A^2 -> H(U1(c, e), U1(k, b))) (A^2 -> H(U1(c, e), U1(d, c)),A^2 -> H(U1(c, e), U1(d, c))) (A^2 -> H(U1(c, e), U1(d, d)),A^2 -> H(U1(c, e), U1(d, d))) ---------------------------------------- (840) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), U1(b, c)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (841) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, c)),A^2 -> H(U1(e, e), U1(b, c))) (A^2 -> H(U1(k, e), U1(b, c)),A^2 -> H(U1(k, e), U1(b, c))) (A^2 -> H(U1(c, e), U1(c, c)),A^2 -> H(U1(c, e), U1(c, c))) (A^2 -> H(U1(c, e), U1(d, c)),A^2 -> H(U1(c, e), U1(d, c))) (A^2 -> H(U1(c, e), U1(b, e)),A^2 -> H(U1(c, e), U1(b, e))) (A^2 -> H(U1(c, e), U1(b, k)),A^2 -> H(U1(c, e), U1(b, k))) ---------------------------------------- (842) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), U1(b, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (843) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, d)),A^2 -> H(U1(e, e), U1(b, d))) (A^2 -> H(U1(k, e), U1(b, d)),A^2 -> H(U1(k, e), U1(b, d))) (A^2 -> H(U1(c, e), U1(c, d)),A^2 -> H(U1(c, e), U1(c, d))) (A^2 -> H(U1(c, e), U1(d, d)),A^2 -> H(U1(c, e), U1(d, d))) (A^2 -> H(U1(c, e), U1(b, k)),A^2 -> H(U1(c, e), U1(b, k))) ---------------------------------------- (844) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), U1(c, c)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (845) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, c)),A^2 -> H(U1(e, e), U1(c, c))) (A^2 -> H(U1(k, e), U1(c, c)),A^2 -> H(U1(k, e), U1(c, c))) (A^2 -> H(U1(c, e), U1(e, c)),A^2 -> H(U1(c, e), U1(e, c))) (A^2 -> H(U1(c, e), U1(k, c)),A^2 -> H(U1(c, e), U1(k, c))) (A^2 -> H(U1(c, e), U1(c, e)),A^2 -> H(U1(c, e), U1(c, e))) (A^2 -> H(U1(c, e), U1(c, k)),A^2 -> H(U1(c, e), U1(c, k))) ---------------------------------------- (846) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), f(e)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (847) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(e)),A^2 -> H(U1(e, e), f(e))) (A^2 -> H(U1(k, e), f(e)),A^2 -> H(U1(k, e), f(e))) (A^2 -> H(U1(c, e), U1(e, e)),A^2 -> H(U1(c, e), U1(e, e))) ---------------------------------------- (848) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), f(k)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (849) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), f(k)),A^2 -> H(U1(e, e), f(k))) (A^2 -> H(U1(k, e), f(k)),A^2 -> H(U1(k, e), f(k))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) ---------------------------------------- (850) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, e), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (851) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, d)),A^2 -> H(U1(e, e), U1(d, d))) (A^2 -> H(U1(k, e), U1(d, d)),A^2 -> H(U1(k, e), U1(d, d))) (A^2 -> H(U1(c, e), U1(k, d)),A^2 -> H(U1(c, e), U1(k, d))) (A^2 -> H(U1(c, e), U1(d, k)),A^2 -> H(U1(c, e), U1(d, k))) ---------------------------------------- (852) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), U1(c, b)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (853) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(c, b)),A^2 -> H(U1(e, k), U1(c, b))) (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(U1(c, k), U1(e, b)),A^2 -> H(U1(c, k), U1(e, b))) (A^2 -> H(U1(c, k), U1(k, b)),A^2 -> H(U1(c, k), U1(k, b))) (A^2 -> H(U1(c, k), U1(c, c)),A^2 -> H(U1(c, k), U1(c, c))) (A^2 -> H(U1(c, k), U1(c, d)),A^2 -> H(U1(c, k), U1(c, d))) ---------------------------------------- (854) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), U1(d, b)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (855) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(d, b)),A^2 -> H(U1(e, k), U1(d, b))) (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(U1(c, k), U1(k, b)),A^2 -> H(U1(c, k), U1(k, b))) (A^2 -> H(U1(c, k), U1(d, c)),A^2 -> H(U1(c, k), U1(d, c))) (A^2 -> H(U1(c, k), U1(d, d)),A^2 -> H(U1(c, k), U1(d, d))) ---------------------------------------- (856) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), U1(b, c)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (857) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(b, c)),A^2 -> H(U1(e, k), U1(b, c))) (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(U1(c, k), U1(c, c)),A^2 -> H(U1(c, k), U1(c, c))) (A^2 -> H(U1(c, k), U1(d, c)),A^2 -> H(U1(c, k), U1(d, c))) (A^2 -> H(U1(c, k), U1(b, e)),A^2 -> H(U1(c, k), U1(b, e))) (A^2 -> H(U1(c, k), U1(b, k)),A^2 -> H(U1(c, k), U1(b, k))) ---------------------------------------- (858) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), U1(b, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (859) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(b, d)),A^2 -> H(U1(e, k), U1(b, d))) (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) (A^2 -> H(U1(c, k), U1(c, d)),A^2 -> H(U1(c, k), U1(c, d))) (A^2 -> H(U1(c, k), U1(d, d)),A^2 -> H(U1(c, k), U1(d, d))) (A^2 -> H(U1(c, k), U1(b, k)),A^2 -> H(U1(c, k), U1(b, k))) ---------------------------------------- (860) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), U1(c, c)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (861) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(c, c)),A^2 -> H(U1(e, k), U1(c, c))) (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(c, k), U1(e, c)),A^2 -> H(U1(c, k), U1(e, c))) (A^2 -> H(U1(c, k), U1(k, c)),A^2 -> H(U1(c, k), U1(k, c))) (A^2 -> H(U1(c, k), U1(c, e)),A^2 -> H(U1(c, k), U1(c, e))) (A^2 -> H(U1(c, k), U1(c, k)),A^2 -> H(U1(c, k), U1(c, k))) ---------------------------------------- (862) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), f(e)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (863) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), f(e)),A^2 -> H(U1(e, k), f(e))) (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) ---------------------------------------- (864) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), f(k)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (865) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), f(k)),A^2 -> H(U1(e, k), f(k))) (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (866) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, k), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (867) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(d, d)),A^2 -> H(U1(e, k), U1(d, d))) (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(c, k), U1(k, d)),A^2 -> H(U1(c, k), U1(k, d))) (A^2 -> H(U1(c, k), U1(d, k)),A^2 -> H(U1(c, k), U1(d, k))) ---------------------------------------- (868) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(e, b)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (869) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(e, b)),A^2 -> H(U1(e, c), U1(e, b))) (A^2 -> H(U1(k, c), U1(e, b)),A^2 -> H(U1(k, c), U1(e, b))) (A^2 -> H(U1(c, e), U1(e, b)),A^2 -> H(U1(c, e), U1(e, b))) (A^2 -> H(U1(c, k), U1(e, b)),A^2 -> H(U1(c, k), U1(e, b))) (A^2 -> H(U1(c, c), b),A^2 -> H(U1(c, c), b)) (A^2 -> H(U1(c, c), U1(e, c)),A^2 -> H(U1(c, c), U1(e, c))) (A^2 -> H(U1(c, c), U1(e, d)),A^2 -> H(U1(c, c), U1(e, d))) ---------------------------------------- (870) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(k, b)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (871) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(k, b)),A^2 -> H(U1(e, c), U1(k, b))) (A^2 -> H(U1(k, c), U1(k, b)),A^2 -> H(U1(k, c), U1(k, b))) (A^2 -> H(U1(c, e), U1(k, b)),A^2 -> H(U1(c, e), U1(k, b))) (A^2 -> H(U1(c, k), U1(k, b)),A^2 -> H(U1(c, k), U1(k, b))) (A^2 -> H(U1(c, c), U1(k, c)),A^2 -> H(U1(c, c), U1(k, c))) (A^2 -> H(U1(c, c), U1(k, d)),A^2 -> H(U1(c, c), U1(k, d))) ---------------------------------------- (872) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(c, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (873) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(c, d)),A^2 -> H(U1(e, c), U1(c, d))) (A^2 -> H(U1(k, c), U1(c, d)),A^2 -> H(U1(k, c), U1(c, d))) (A^2 -> H(U1(c, e), U1(c, d)),A^2 -> H(U1(c, e), U1(c, d))) (A^2 -> H(U1(c, k), U1(c, d)),A^2 -> H(U1(c, k), U1(c, d))) (A^2 -> H(U1(c, c), U1(e, d)),A^2 -> H(U1(c, c), U1(e, d))) (A^2 -> H(U1(c, c), U1(k, d)),A^2 -> H(U1(c, c), U1(k, d))) (A^2 -> H(U1(c, c), U1(c, k)),A^2 -> H(U1(c, c), U1(c, k))) ---------------------------------------- (874) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(d, c)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (875) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(d, c)),A^2 -> H(U1(e, c), U1(d, c))) (A^2 -> H(U1(k, c), U1(d, c)),A^2 -> H(U1(k, c), U1(d, c))) (A^2 -> H(U1(c, e), U1(d, c)),A^2 -> H(U1(c, e), U1(d, c))) (A^2 -> H(U1(c, k), U1(d, c)),A^2 -> H(U1(c, k), U1(d, c))) (A^2 -> H(U1(c, c), U1(k, c)),A^2 -> H(U1(c, c), U1(k, c))) (A^2 -> H(U1(c, c), U1(d, e)),A^2 -> H(U1(c, c), U1(d, e))) (A^2 -> H(U1(c, c), U1(d, k)),A^2 -> H(U1(c, c), U1(d, k))) ---------------------------------------- (876) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(b, e)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (877) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(b, e)),A^2 -> H(U1(e, c), U1(b, e))) (A^2 -> H(U1(k, c), U1(b, e)),A^2 -> H(U1(k, c), U1(b, e))) (A^2 -> H(U1(c, e), U1(b, e)),A^2 -> H(U1(c, e), U1(b, e))) (A^2 -> H(U1(c, k), U1(b, e)),A^2 -> H(U1(c, k), U1(b, e))) (A^2 -> H(U1(c, c), U1(c, e)),A^2 -> H(U1(c, c), U1(c, e))) (A^2 -> H(U1(c, c), U1(d, e)),A^2 -> H(U1(c, c), U1(d, e))) ---------------------------------------- (878) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(b, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (879) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(b, k)),A^2 -> H(U1(e, c), U1(b, k))) (A^2 -> H(U1(k, c), U1(b, k)),A^2 -> H(U1(k, c), U1(b, k))) (A^2 -> H(U1(c, e), U1(b, k)),A^2 -> H(U1(c, e), U1(b, k))) (A^2 -> H(U1(c, k), U1(b, k)),A^2 -> H(U1(c, k), U1(b, k))) (A^2 -> H(U1(c, c), U1(c, k)),A^2 -> H(U1(c, c), U1(c, k))) (A^2 -> H(U1(c, c), U1(d, k)),A^2 -> H(U1(c, c), U1(d, k))) ---------------------------------------- (880) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (881) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(e, e)),A^2 -> H(U1(e, c), U1(e, e))) (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) (A^2 -> H(U1(c, e), U1(e, e)),A^2 -> H(U1(c, e), U1(e, e))) (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) (A^2 -> H(U1(c, c), e),A^2 -> H(U1(c, c), e)) ---------------------------------------- (882) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (883) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (884) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(k, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (885) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(k, d)),A^2 -> H(U1(e, c), U1(k, d))) (A^2 -> H(U1(k, c), U1(k, d)),A^2 -> H(U1(k, c), U1(k, d))) (A^2 -> H(U1(c, e), U1(k, d)),A^2 -> H(U1(c, e), U1(k, d))) (A^2 -> H(U1(c, k), U1(k, d)),A^2 -> H(U1(c, k), U1(k, d))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) ---------------------------------------- (886) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(d, k)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (887) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(d, k)),A^2 -> H(U1(e, c), U1(d, k))) (A^2 -> H(U1(k, c), U1(d, k)),A^2 -> H(U1(k, c), U1(d, k))) (A^2 -> H(U1(c, e), U1(d, k)),A^2 -> H(U1(c, e), U1(d, k))) (A^2 -> H(U1(c, k), U1(d, k)),A^2 -> H(U1(c, k), U1(d, k))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) ---------------------------------------- (888) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(e, U1(b, b)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (889) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, b)),A^2 -> H(e, U1(c, b))) (A^2 -> H(e, U1(d, b)),A^2 -> H(e, U1(d, b))) (A^2 -> H(e, U1(b, c)),A^2 -> H(e, U1(b, c))) (A^2 -> H(e, U1(b, d)),A^2 -> H(e, U1(b, d))) ---------------------------------------- (890) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(e, f(c)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (891) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, c)),A^2 -> H(e, U1(c, c))) (A^2 -> H(e, f(e)),A^2 -> H(e, f(e))) (A^2 -> H(e, f(k)),A^2 -> H(e, f(k))) ---------------------------------------- (892) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(e, f(d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (893) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, d)),A^2 -> H(e, U1(d, d))) (A^2 -> H(e, f(k)),A^2 -> H(e, f(k))) ---------------------------------------- (894) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), U1(c, b)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (895) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, b)),A^2 -> H(e, U1(c, b))) (A^2 -> H(U1(e, e), U1(e, b)),A^2 -> H(U1(e, e), U1(e, b))) (A^2 -> H(U1(e, e), U1(k, b)),A^2 -> H(U1(e, e), U1(k, b))) (A^2 -> H(U1(e, e), U1(c, c)),A^2 -> H(U1(e, e), U1(c, c))) (A^2 -> H(U1(e, e), U1(c, d)),A^2 -> H(U1(e, e), U1(c, d))) ---------------------------------------- (896) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), U1(d, b)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (897) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, b)),A^2 -> H(e, U1(d, b))) (A^2 -> H(U1(e, e), U1(k, b)),A^2 -> H(U1(e, e), U1(k, b))) (A^2 -> H(U1(e, e), U1(d, c)),A^2 -> H(U1(e, e), U1(d, c))) (A^2 -> H(U1(e, e), U1(d, d)),A^2 -> H(U1(e, e), U1(d, d))) ---------------------------------------- (898) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), U1(b, c)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (899) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, c)),A^2 -> H(e, U1(b, c))) (A^2 -> H(U1(e, e), U1(c, c)),A^2 -> H(U1(e, e), U1(c, c))) (A^2 -> H(U1(e, e), U1(d, c)),A^2 -> H(U1(e, e), U1(d, c))) (A^2 -> H(U1(e, e), U1(b, e)),A^2 -> H(U1(e, e), U1(b, e))) (A^2 -> H(U1(e, e), U1(b, k)),A^2 -> H(U1(e, e), U1(b, k))) ---------------------------------------- (900) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), U1(b, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (901) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, d)),A^2 -> H(e, U1(b, d))) (A^2 -> H(U1(e, e), U1(c, d)),A^2 -> H(U1(e, e), U1(c, d))) (A^2 -> H(U1(e, e), U1(d, d)),A^2 -> H(U1(e, e), U1(d, d))) (A^2 -> H(U1(e, e), U1(b, k)),A^2 -> H(U1(e, e), U1(b, k))) ---------------------------------------- (902) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), U1(c, c)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (903) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, c)),A^2 -> H(e, U1(c, c))) (A^2 -> H(U1(e, e), U1(e, c)),A^2 -> H(U1(e, e), U1(e, c))) (A^2 -> H(U1(e, e), U1(k, c)),A^2 -> H(U1(e, e), U1(k, c))) (A^2 -> H(U1(e, e), U1(c, e)),A^2 -> H(U1(e, e), U1(c, e))) (A^2 -> H(U1(e, e), U1(c, k)),A^2 -> H(U1(e, e), U1(c, k))) ---------------------------------------- (904) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), f(e)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (905) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(e)),A^2 -> H(e, f(e))) (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) ---------------------------------------- (906) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), f(k)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (907) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(k)),A^2 -> H(e, f(k))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (908) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, e), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (909) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, d)),A^2 -> H(e, U1(d, d))) (A^2 -> H(U1(e, e), U1(k, d)),A^2 -> H(U1(e, e), U1(k, d))) (A^2 -> H(U1(e, e), U1(d, k)),A^2 -> H(U1(e, e), U1(d, k))) ---------------------------------------- (910) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(e, b)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (911) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, b)),A^2 -> H(U1(e, e), U1(e, b))) (A^2 -> H(f(e), b),A^2 -> H(f(e), b)) (A^2 -> H(f(e), U1(e, c)),A^2 -> H(f(e), U1(e, c))) (A^2 -> H(f(e), U1(e, d)),A^2 -> H(f(e), U1(e, d))) ---------------------------------------- (912) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(k, b)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (913) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, b)),A^2 -> H(U1(e, e), U1(k, b))) (A^2 -> H(f(e), U1(k, c)),A^2 -> H(f(e), U1(k, c))) (A^2 -> H(f(e), U1(k, d)),A^2 -> H(f(e), U1(k, d))) ---------------------------------------- (914) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(c, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (915) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, d)),A^2 -> H(U1(e, e), U1(c, d))) (A^2 -> H(f(e), U1(e, d)),A^2 -> H(f(e), U1(e, d))) (A^2 -> H(f(e), U1(k, d)),A^2 -> H(f(e), U1(k, d))) (A^2 -> H(f(e), U1(c, k)),A^2 -> H(f(e), U1(c, k))) ---------------------------------------- (916) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(d, c)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (917) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, c)),A^2 -> H(U1(e, e), U1(d, c))) (A^2 -> H(f(e), U1(k, c)),A^2 -> H(f(e), U1(k, c))) (A^2 -> H(f(e), U1(d, e)),A^2 -> H(f(e), U1(d, e))) (A^2 -> H(f(e), U1(d, k)),A^2 -> H(f(e), U1(d, k))) ---------------------------------------- (918) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(b, e)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (919) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, e)),A^2 -> H(U1(e, e), U1(b, e))) (A^2 -> H(f(e), U1(c, e)),A^2 -> H(f(e), U1(c, e))) (A^2 -> H(f(e), U1(d, e)),A^2 -> H(f(e), U1(d, e))) ---------------------------------------- (920) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(b, k)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (921) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, k)),A^2 -> H(U1(e, e), U1(b, k))) (A^2 -> H(f(e), U1(c, k)),A^2 -> H(f(e), U1(c, k))) (A^2 -> H(f(e), U1(d, k)),A^2 -> H(f(e), U1(d, k))) ---------------------------------------- (922) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(e, c)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (923) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, c)),A^2 -> H(U1(e, e), U1(e, c))) (A^2 -> H(f(e), c),A^2 -> H(f(e), c)) (A^2 -> H(f(e), U1(e, e)),A^2 -> H(f(e), U1(e, e))) (A^2 -> H(f(e), U1(e, k)),A^2 -> H(f(e), U1(e, k))) ---------------------------------------- (924) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(k, c)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (925) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, c)),A^2 -> H(U1(e, e), U1(k, c))) (A^2 -> H(f(e), U1(k, e)),A^2 -> H(f(e), U1(k, e))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) ---------------------------------------- (926) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(c, e)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (927) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, e)),A^2 -> H(U1(e, e), U1(c, e))) (A^2 -> H(f(e), U1(e, e)),A^2 -> H(f(e), U1(e, e))) (A^2 -> H(f(e), U1(k, e)),A^2 -> H(f(e), U1(k, e))) ---------------------------------------- (928) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(c, k)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (929) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, k)),A^2 -> H(U1(e, e), U1(c, k))) (A^2 -> H(f(e), U1(e, k)),A^2 -> H(f(e), U1(e, k))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) ---------------------------------------- (930) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (931) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (932) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(k, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (933) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, d)),A^2 -> H(U1(e, e), U1(k, d))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) ---------------------------------------- (934) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(d, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (935) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, k)),A^2 -> H(U1(e, e), U1(d, k))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) ---------------------------------------- (936) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (937) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) ---------------------------------------- (938) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (939) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) ---------------------------------------- (940) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (941) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) ---------------------------------------- (942) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (943) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) ---------------------------------------- (944) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (945) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) ---------------------------------------- (946) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (947) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) ---------------------------------------- (948) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (949) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (950) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (951) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) ---------------------------------------- (952) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(e, b)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (953) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(f(k), b),A^2 -> H(f(k), b)) (A^2 -> H(f(k), U1(e, c)),A^2 -> H(f(k), U1(e, c))) (A^2 -> H(f(k), U1(e, d)),A^2 -> H(f(k), U1(e, d))) ---------------------------------------- (954) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(k, b)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (955) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(f(k), U1(k, c)),A^2 -> H(f(k), U1(k, c))) (A^2 -> H(f(k), U1(k, d)),A^2 -> H(f(k), U1(k, d))) ---------------------------------------- (956) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(c, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (957) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(f(k), U1(e, d)),A^2 -> H(f(k), U1(e, d))) (A^2 -> H(f(k), U1(k, d)),A^2 -> H(f(k), U1(k, d))) (A^2 -> H(f(k), U1(c, k)),A^2 -> H(f(k), U1(c, k))) ---------------------------------------- (958) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(d, c)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (959) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(f(k), U1(k, c)),A^2 -> H(f(k), U1(k, c))) (A^2 -> H(f(k), U1(d, e)),A^2 -> H(f(k), U1(d, e))) (A^2 -> H(f(k), U1(d, k)),A^2 -> H(f(k), U1(d, k))) ---------------------------------------- (960) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(b, e)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (961) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(f(k), U1(c, e)),A^2 -> H(f(k), U1(c, e))) (A^2 -> H(f(k), U1(d, e)),A^2 -> H(f(k), U1(d, e))) ---------------------------------------- (962) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(b, k)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (963) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) (A^2 -> H(f(k), U1(c, k)),A^2 -> H(f(k), U1(c, k))) (A^2 -> H(f(k), U1(d, k)),A^2 -> H(f(k), U1(d, k))) ---------------------------------------- (964) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(e, c)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (965) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(f(k), c),A^2 -> H(f(k), c)) (A^2 -> H(f(k), U1(e, e)),A^2 -> H(f(k), U1(e, e))) (A^2 -> H(f(k), U1(e, k)),A^2 -> H(f(k), U1(e, k))) ---------------------------------------- (966) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(k, c)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (967) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(f(k), U1(k, e)),A^2 -> H(f(k), U1(k, e))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (968) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(c, e)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (969) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(f(k), U1(e, e)),A^2 -> H(f(k), U1(e, e))) (A^2 -> H(f(k), U1(k, e)),A^2 -> H(f(k), U1(k, e))) ---------------------------------------- (970) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(c, k)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (971) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) (A^2 -> H(f(k), U1(e, k)),A^2 -> H(f(k), U1(e, k))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (972) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(e, e)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (973) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(f(k), e),A^2 -> H(f(k), e)) ---------------------------------------- (974) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(k, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (975) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (976) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(d, k)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (977) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (978) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), b) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (979) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), b),A^2 -> H(U1(c, c), b)) (A^2 -> H(f(e), b),A^2 -> H(f(e), b)) (A^2 -> H(f(k), b),A^2 -> H(f(k), b)) (A^2 -> H(f(c), c),A^2 -> H(f(c), c)) (A^2 -> H(f(c), d),A^2 -> H(f(c), d)) ---------------------------------------- (980) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(e, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (981) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, d)),A^2 -> H(U1(c, c), U1(e, d))) (A^2 -> H(f(e), U1(e, d)),A^2 -> H(f(e), U1(e, d))) (A^2 -> H(f(k), U1(e, d)),A^2 -> H(f(k), U1(e, d))) (A^2 -> H(f(c), d),A^2 -> H(f(c), d)) (A^2 -> H(f(c), U1(e, k)),A^2 -> H(f(c), U1(e, k))) ---------------------------------------- (982) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(d, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (983) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, e)),A^2 -> H(U1(c, c), U1(d, e))) (A^2 -> H(f(e), U1(d, e)),A^2 -> H(f(e), U1(d, e))) (A^2 -> H(f(k), U1(d, e)),A^2 -> H(f(k), U1(d, e))) (A^2 -> H(f(c), U1(k, e)),A^2 -> H(f(c), U1(k, e))) ---------------------------------------- (984) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (985) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (986) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), U1(b, b)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (987) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) ---------------------------------------- (988) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), f(c)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (989) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) ---------------------------------------- (990) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, k), f(d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (991) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) ---------------------------------------- (992) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), U1(c, b)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (993) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(U1(k, d), U1(e, b)),A^2 -> H(U1(k, d), U1(e, b))) (A^2 -> H(U1(k, d), U1(k, b)),A^2 -> H(U1(k, d), U1(k, b))) (A^2 -> H(U1(k, d), U1(c, c)),A^2 -> H(U1(k, d), U1(c, c))) (A^2 -> H(U1(k, d), U1(c, d)),A^2 -> H(U1(k, d), U1(c, d))) ---------------------------------------- (994) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), U1(d, b)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (995) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(U1(k, d), U1(k, b)),A^2 -> H(U1(k, d), U1(k, b))) (A^2 -> H(U1(k, d), U1(d, c)),A^2 -> H(U1(k, d), U1(d, c))) (A^2 -> H(U1(k, d), U1(d, d)),A^2 -> H(U1(k, d), U1(d, d))) ---------------------------------------- (996) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), U1(b, c)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (997) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(U1(k, d), U1(c, c)),A^2 -> H(U1(k, d), U1(c, c))) (A^2 -> H(U1(k, d), U1(d, c)),A^2 -> H(U1(k, d), U1(d, c))) (A^2 -> H(U1(k, d), U1(b, e)),A^2 -> H(U1(k, d), U1(b, e))) (A^2 -> H(U1(k, d), U1(b, k)),A^2 -> H(U1(k, d), U1(b, k))) ---------------------------------------- (998) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), U1(b, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (999) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) (A^2 -> H(U1(k, d), U1(c, d)),A^2 -> H(U1(k, d), U1(c, d))) (A^2 -> H(U1(k, d), U1(d, d)),A^2 -> H(U1(k, d), U1(d, d))) (A^2 -> H(U1(k, d), U1(b, k)),A^2 -> H(U1(k, d), U1(b, k))) ---------------------------------------- (1000) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), U1(c, c)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1001) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, d), U1(e, c)),A^2 -> H(U1(k, d), U1(e, c))) (A^2 -> H(U1(k, d), U1(k, c)),A^2 -> H(U1(k, d), U1(k, c))) (A^2 -> H(U1(k, d), U1(c, e)),A^2 -> H(U1(k, d), U1(c, e))) (A^2 -> H(U1(k, d), U1(c, k)),A^2 -> H(U1(k, d), U1(c, k))) ---------------------------------------- (1002) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), f(e)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1003) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(U1(k, d), U1(e, e)),A^2 -> H(U1(k, d), U1(e, e))) ---------------------------------------- (1004) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), f(k)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1005) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) ---------------------------------------- (1006) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1007) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(k, d), U1(k, d)),A^2 -> H(U1(k, d), U1(k, d))) (A^2 -> H(U1(k, d), U1(d, k)),A^2 -> H(U1(k, d), U1(d, k))) ---------------------------------------- (1008) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, b)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1009) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, b)),A^2 -> H(U1(k, k), U1(c, b))) (A^2 -> H(U1(d, k), U1(e, b)),A^2 -> H(U1(d, k), U1(e, b))) (A^2 -> H(U1(d, k), U1(k, b)),A^2 -> H(U1(d, k), U1(k, b))) (A^2 -> H(U1(d, k), U1(c, c)),A^2 -> H(U1(d, k), U1(c, c))) (A^2 -> H(U1(d, k), U1(c, d)),A^2 -> H(U1(d, k), U1(c, d))) ---------------------------------------- (1010) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), U1(d, b)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1011) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, b)),A^2 -> H(U1(k, k), U1(d, b))) (A^2 -> H(U1(d, k), U1(k, b)),A^2 -> H(U1(d, k), U1(k, b))) (A^2 -> H(U1(d, k), U1(d, c)),A^2 -> H(U1(d, k), U1(d, c))) (A^2 -> H(U1(d, k), U1(d, d)),A^2 -> H(U1(d, k), U1(d, d))) ---------------------------------------- (1012) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), U1(b, c)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1013) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, c)),A^2 -> H(U1(k, k), U1(b, c))) (A^2 -> H(U1(d, k), U1(c, c)),A^2 -> H(U1(d, k), U1(c, c))) (A^2 -> H(U1(d, k), U1(d, c)),A^2 -> H(U1(d, k), U1(d, c))) (A^2 -> H(U1(d, k), U1(b, e)),A^2 -> H(U1(d, k), U1(b, e))) (A^2 -> H(U1(d, k), U1(b, k)),A^2 -> H(U1(d, k), U1(b, k))) ---------------------------------------- (1014) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), U1(b, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1015) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, d)),A^2 -> H(U1(k, k), U1(b, d))) (A^2 -> H(U1(d, k), U1(c, d)),A^2 -> H(U1(d, k), U1(c, d))) (A^2 -> H(U1(d, k), U1(d, d)),A^2 -> H(U1(d, k), U1(d, d))) (A^2 -> H(U1(d, k), U1(b, k)),A^2 -> H(U1(d, k), U1(b, k))) ---------------------------------------- (1016) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), U1(c, c)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1017) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(d, k), U1(e, c)),A^2 -> H(U1(d, k), U1(e, c))) (A^2 -> H(U1(d, k), U1(k, c)),A^2 -> H(U1(d, k), U1(k, c))) (A^2 -> H(U1(d, k), U1(c, e)),A^2 -> H(U1(d, k), U1(c, e))) (A^2 -> H(U1(d, k), U1(c, k)),A^2 -> H(U1(d, k), U1(c, k))) ---------------------------------------- (1018) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), f(e)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1019) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(e)),A^2 -> H(U1(k, k), f(e))) (A^2 -> H(U1(d, k), U1(e, e)),A^2 -> H(U1(d, k), U1(e, e))) ---------------------------------------- (1020) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), f(k)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1021) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), f(k)),A^2 -> H(U1(k, k), f(k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1022) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, k), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1023) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(d, k), U1(k, d)),A^2 -> H(U1(d, k), U1(k, d))) (A^2 -> H(U1(d, k), U1(d, k)),A^2 -> H(U1(d, k), U1(d, k))) ---------------------------------------- (1024) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(e, b)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1025) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(e, b)),A^2 -> H(U1(k, d), U1(e, b))) (A^2 -> H(U1(d, k), U1(e, b)),A^2 -> H(U1(d, k), U1(e, b))) (A^2 -> H(U1(d, d), b),A^2 -> H(U1(d, d), b)) (A^2 -> H(U1(d, d), U1(e, c)),A^2 -> H(U1(d, d), U1(e, c))) (A^2 -> H(U1(d, d), U1(e, d)),A^2 -> H(U1(d, d), U1(e, d))) ---------------------------------------- (1026) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(k, b)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1027) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(k, b)),A^2 -> H(U1(k, d), U1(k, b))) (A^2 -> H(U1(d, k), U1(k, b)),A^2 -> H(U1(d, k), U1(k, b))) (A^2 -> H(U1(d, d), U1(k, c)),A^2 -> H(U1(d, d), U1(k, c))) (A^2 -> H(U1(d, d), U1(k, d)),A^2 -> H(U1(d, d), U1(k, d))) ---------------------------------------- (1028) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(c, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1029) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(c, d)),A^2 -> H(U1(k, d), U1(c, d))) (A^2 -> H(U1(d, k), U1(c, d)),A^2 -> H(U1(d, k), U1(c, d))) (A^2 -> H(U1(d, d), U1(e, d)),A^2 -> H(U1(d, d), U1(e, d))) (A^2 -> H(U1(d, d), U1(k, d)),A^2 -> H(U1(d, d), U1(k, d))) (A^2 -> H(U1(d, d), U1(c, k)),A^2 -> H(U1(d, d), U1(c, k))) ---------------------------------------- (1030) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(d, c)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1031) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(d, c)),A^2 -> H(U1(k, d), U1(d, c))) (A^2 -> H(U1(d, k), U1(d, c)),A^2 -> H(U1(d, k), U1(d, c))) (A^2 -> H(U1(d, d), U1(k, c)),A^2 -> H(U1(d, d), U1(k, c))) (A^2 -> H(U1(d, d), U1(d, e)),A^2 -> H(U1(d, d), U1(d, e))) (A^2 -> H(U1(d, d), U1(d, k)),A^2 -> H(U1(d, d), U1(d, k))) ---------------------------------------- (1032) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(b, e)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1033) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(b, e)),A^2 -> H(U1(k, d), U1(b, e))) (A^2 -> H(U1(d, k), U1(b, e)),A^2 -> H(U1(d, k), U1(b, e))) (A^2 -> H(U1(d, d), U1(c, e)),A^2 -> H(U1(d, d), U1(c, e))) (A^2 -> H(U1(d, d), U1(d, e)),A^2 -> H(U1(d, d), U1(d, e))) ---------------------------------------- (1034) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(b, k)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1035) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(b, k)),A^2 -> H(U1(k, d), U1(b, k))) (A^2 -> H(U1(d, k), U1(b, k)),A^2 -> H(U1(d, k), U1(b, k))) (A^2 -> H(U1(d, d), U1(c, k)),A^2 -> H(U1(d, d), U1(c, k))) (A^2 -> H(U1(d, d), U1(d, k)),A^2 -> H(U1(d, d), U1(d, k))) ---------------------------------------- (1036) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(e, c)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1037) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(e, c)),A^2 -> H(U1(k, d), U1(e, c))) (A^2 -> H(U1(d, k), U1(e, c)),A^2 -> H(U1(d, k), U1(e, c))) (A^2 -> H(U1(d, d), c),A^2 -> H(U1(d, d), c)) (A^2 -> H(U1(d, d), U1(e, e)),A^2 -> H(U1(d, d), U1(e, e))) (A^2 -> H(U1(d, d), U1(e, k)),A^2 -> H(U1(d, d), U1(e, k))) ---------------------------------------- (1038) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(k, c)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1039) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(k, c)),A^2 -> H(U1(k, d), U1(k, c))) (A^2 -> H(U1(d, k), U1(k, c)),A^2 -> H(U1(d, k), U1(k, c))) (A^2 -> H(U1(d, d), U1(k, e)),A^2 -> H(U1(d, d), U1(k, e))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (1040) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(c, e)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1041) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(c, e)),A^2 -> H(U1(k, d), U1(c, e))) (A^2 -> H(U1(d, k), U1(c, e)),A^2 -> H(U1(d, k), U1(c, e))) (A^2 -> H(U1(d, d), U1(e, e)),A^2 -> H(U1(d, d), U1(e, e))) (A^2 -> H(U1(d, d), U1(k, e)),A^2 -> H(U1(d, d), U1(k, e))) ---------------------------------------- (1042) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(c, k)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1043) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(c, k)),A^2 -> H(U1(k, d), U1(c, k))) (A^2 -> H(U1(d, k), U1(c, k)),A^2 -> H(U1(d, k), U1(c, k))) (A^2 -> H(U1(d, d), U1(e, k)),A^2 -> H(U1(d, d), U1(e, k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (1044) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(e, e)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1045) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(e, e)),A^2 -> H(U1(k, d), U1(e, e))) (A^2 -> H(U1(d, k), U1(e, e)),A^2 -> H(U1(d, k), U1(e, e))) (A^2 -> H(U1(d, d), e),A^2 -> H(U1(d, d), e)) ---------------------------------------- (1046) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1047) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1048) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), b) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1049) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), b),A^2 -> H(U1(d, d), b)) (A^2 -> H(f(k), b),A^2 -> H(f(k), b)) (A^2 -> H(f(d), c),A^2 -> H(f(d), c)) (A^2 -> H(f(d), d),A^2 -> H(f(d), d)) ---------------------------------------- (1050) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1051) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(e, d)),A^2 -> H(U1(d, d), U1(e, d))) (A^2 -> H(f(k), U1(e, d)),A^2 -> H(f(k), U1(e, d))) (A^2 -> H(f(d), d),A^2 -> H(f(d), d)) (A^2 -> H(f(d), U1(e, k)),A^2 -> H(f(d), U1(e, k))) ---------------------------------------- (1052) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(d, e)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1053) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(d, e)),A^2 -> H(U1(d, d), U1(d, e))) (A^2 -> H(f(k), U1(d, e)),A^2 -> H(f(k), U1(d, e))) (A^2 -> H(f(d), U1(k, e)),A^2 -> H(f(d), U1(k, e))) ---------------------------------------- (1054) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), c) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1055) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), c),A^2 -> H(U1(d, d), c)) (A^2 -> H(f(k), c),A^2 -> H(f(k), c)) (A^2 -> H(f(d), e),A^2 -> H(f(d), e)) (A^2 -> H(f(d), k),A^2 -> H(f(d), k)) ---------------------------------------- (1056) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(e, k)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1057) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(e, k)),A^2 -> H(U1(d, d), U1(e, k))) (A^2 -> H(f(k), U1(e, k)),A^2 -> H(f(k), U1(e, k))) (A^2 -> H(f(d), k),A^2 -> H(f(d), k)) ---------------------------------------- (1058) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(k, e)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1059) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(k, e)),A^2 -> H(U1(d, d), U1(k, e))) (A^2 -> H(f(k), U1(k, e)),A^2 -> H(f(k), U1(k, e))) ---------------------------------------- (1060) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), e) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1061) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), e),A^2 -> H(U1(d, d), e)) (A^2 -> H(f(k), e),A^2 -> H(f(k), e)) ---------------------------------------- (1062) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), d) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1063) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), d),A^2 -> H(U1(a, a), d)) (A^2 -> H(f(c), d),A^2 -> H(f(c), d)) (A^2 -> H(f(d), d),A^2 -> H(f(d), d)) (A^2 -> H(f(a), k),A^2 -> H(f(a), k)) ---------------------------------------- (1064) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(e, c)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1065) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(e, c)),A^2 -> H(U1(e, c), U1(e, c))) (A^2 -> H(U1(k, c), U1(e, c)),A^2 -> H(U1(k, c), U1(e, c))) (A^2 -> H(U1(c, e), U1(e, c)),A^2 -> H(U1(c, e), U1(e, c))) (A^2 -> H(U1(c, k), U1(e, c)),A^2 -> H(U1(c, k), U1(e, c))) (A^2 -> H(U1(c, c), c),A^2 -> H(U1(c, c), c)) (A^2 -> H(U1(c, c), U1(e, e)),A^2 -> H(U1(c, c), U1(e, e))) (A^2 -> H(U1(c, c), U1(e, k)),A^2 -> H(U1(c, c), U1(e, k))) ---------------------------------------- (1066) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), c) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1067) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), c),A^2 -> H(U1(c, c), c)) (A^2 -> H(f(e), c),A^2 -> H(f(e), c)) (A^2 -> H(f(k), c),A^2 -> H(f(k), c)) (A^2 -> H(f(c), e),A^2 -> H(f(c), e)) (A^2 -> H(f(c), k),A^2 -> H(f(c), k)) ---------------------------------------- (1068) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(e, k)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1069) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, k)),A^2 -> H(U1(c, c), U1(e, k))) (A^2 -> H(f(e), U1(e, k)),A^2 -> H(f(e), U1(e, k))) (A^2 -> H(f(k), U1(e, k)),A^2 -> H(f(k), U1(e, k))) (A^2 -> H(f(c), k),A^2 -> H(f(c), k)) ---------------------------------------- (1070) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(k, c)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1071) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(k, c)),A^2 -> H(U1(e, c), U1(k, c))) (A^2 -> H(U1(k, c), U1(k, c)),A^2 -> H(U1(k, c), U1(k, c))) (A^2 -> H(U1(c, e), U1(k, c)),A^2 -> H(U1(c, e), U1(k, c))) (A^2 -> H(U1(c, k), U1(k, c)),A^2 -> H(U1(c, k), U1(k, c))) (A^2 -> H(U1(c, c), U1(k, e)),A^2 -> H(U1(c, c), U1(k, e))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) ---------------------------------------- (1072) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(k, e)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1073) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, e)),A^2 -> H(U1(c, c), U1(k, e))) (A^2 -> H(f(e), U1(k, e)),A^2 -> H(f(e), U1(k, e))) (A^2 -> H(f(k), U1(k, e)),A^2 -> H(f(k), U1(k, e))) ---------------------------------------- (1074) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), U1(k, k)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1075) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) (A^2 -> H(f(e), U1(k, k)),A^2 -> H(f(e), U1(k, k))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (1076) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(c, e)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1077) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(c, e)),A^2 -> H(U1(e, c), U1(c, e))) (A^2 -> H(U1(k, c), U1(c, e)),A^2 -> H(U1(k, c), U1(c, e))) (A^2 -> H(U1(c, e), U1(c, e)),A^2 -> H(U1(c, e), U1(c, e))) (A^2 -> H(U1(c, k), U1(c, e)),A^2 -> H(U1(c, k), U1(c, e))) (A^2 -> H(U1(c, c), U1(e, e)),A^2 -> H(U1(c, c), U1(e, e))) (A^2 -> H(U1(c, c), U1(k, e)),A^2 -> H(U1(c, c), U1(k, e))) ---------------------------------------- (1078) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, c), U1(c, k)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1079) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(c, k)),A^2 -> H(U1(e, c), U1(c, k))) (A^2 -> H(U1(k, c), U1(c, k)),A^2 -> H(U1(k, c), U1(c, k))) (A^2 -> H(U1(c, e), U1(c, k)),A^2 -> H(U1(c, e), U1(c, k))) (A^2 -> H(U1(c, k), U1(c, k)),A^2 -> H(U1(c, k), U1(c, k))) (A^2 -> H(U1(c, c), U1(e, k)),A^2 -> H(U1(c, c), U1(e, k))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) ---------------------------------------- (1080) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(a), k) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1081) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(a), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(a, a), k),A^2 -> H(U1(a, a), k)) (A^2 -> H(f(c), k),A^2 -> H(f(c), k)) (A^2 -> H(f(d), k),A^2 -> H(f(d), k)) ---------------------------------------- (1082) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(e), U1(e, e)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1083) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(f(e), e),A^2 -> H(f(e), e)) ---------------------------------------- (1084) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(c), e) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1085) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), e),A^2 -> H(U1(c, c), e)) (A^2 -> H(f(e), e),A^2 -> H(f(e), e)) (A^2 -> H(f(k), e),A^2 -> H(f(k), e)) ---------------------------------------- (1086) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(k, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1087) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(k, d)),A^2 -> H(U1(k, d), U1(k, d))) (A^2 -> H(U1(d, k), U1(k, d)),A^2 -> H(U1(d, k), U1(k, d))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (1088) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(f(d), U1(k, k)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1089) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) (A^2 -> H(f(k), U1(k, k)),A^2 -> H(f(k), U1(k, k))) ---------------------------------------- (1090) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(d, d), U1(d, k)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1091) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(d, k)),A^2 -> H(U1(k, d), U1(d, k))) (A^2 -> H(U1(d, k), U1(d, k)),A^2 -> H(U1(d, k), U1(d, k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (1092) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(d, U1(b, b)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1093) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, b)),A^2 -> H(k, U1(b, b))) (A^2 -> H(d, U1(c, b)),A^2 -> H(d, U1(c, b))) (A^2 -> H(d, U1(d, b)),A^2 -> H(d, U1(d, b))) (A^2 -> H(d, U1(b, c)),A^2 -> H(d, U1(b, c))) (A^2 -> H(d, U1(b, d)),A^2 -> H(d, U1(b, d))) ---------------------------------------- (1094) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(d, f(c)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1095) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(c)),A^2 -> H(k, f(c))) (A^2 -> H(d, U1(c, c)),A^2 -> H(d, U1(c, c))) (A^2 -> H(d, f(e)),A^2 -> H(d, f(e))) (A^2 -> H(d, f(k)),A^2 -> H(d, f(k))) ---------------------------------------- (1096) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(d, f(d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1097) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(d)),A^2 -> H(k, f(d))) (A^2 -> H(d, U1(d, d)),A^2 -> H(d, U1(d, d))) (A^2 -> H(d, f(k)),A^2 -> H(d, f(k))) ---------------------------------------- (1098) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(c, b)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1099) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, b)),A^2 -> H(c, U1(c, b))) (A^2 -> H(d, U1(c, b)),A^2 -> H(d, U1(c, b))) (A^2 -> H(a, U1(e, b)),A^2 -> H(a, U1(e, b))) (A^2 -> H(a, U1(k, b)),A^2 -> H(a, U1(k, b))) (A^2 -> H(a, U1(c, c)),A^2 -> H(a, U1(c, c))) (A^2 -> H(a, U1(c, d)),A^2 -> H(a, U1(c, d))) ---------------------------------------- (1100) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(d, b)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1101) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, b)),A^2 -> H(c, U1(d, b))) (A^2 -> H(d, U1(d, b)),A^2 -> H(d, U1(d, b))) (A^2 -> H(a, U1(k, b)),A^2 -> H(a, U1(k, b))) (A^2 -> H(a, U1(d, c)),A^2 -> H(a, U1(d, c))) (A^2 -> H(a, U1(d, d)),A^2 -> H(a, U1(d, d))) ---------------------------------------- (1102) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(b, c)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, c)),A^2 -> H(c, U1(b, c))) (A^2 -> H(d, U1(b, c)),A^2 -> H(d, U1(b, c))) (A^2 -> H(a, U1(c, c)),A^2 -> H(a, U1(c, c))) (A^2 -> H(a, U1(d, c)),A^2 -> H(a, U1(d, c))) (A^2 -> H(a, U1(b, e)),A^2 -> H(a, U1(b, e))) (A^2 -> H(a, U1(b, k)),A^2 -> H(a, U1(b, k))) ---------------------------------------- (1104) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(b, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1105) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, d)),A^2 -> H(c, U1(b, d))) (A^2 -> H(d, U1(b, d)),A^2 -> H(d, U1(b, d))) (A^2 -> H(a, U1(c, d)),A^2 -> H(a, U1(c, d))) (A^2 -> H(a, U1(d, d)),A^2 -> H(a, U1(d, d))) (A^2 -> H(a, U1(b, k)),A^2 -> H(a, U1(b, k))) ---------------------------------------- (1106) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(c, c)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, c)),A^2 -> H(c, U1(c, c))) (A^2 -> H(d, U1(c, c)),A^2 -> H(d, U1(c, c))) (A^2 -> H(a, U1(e, c)),A^2 -> H(a, U1(e, c))) (A^2 -> H(a, U1(k, c)),A^2 -> H(a, U1(k, c))) (A^2 -> H(a, U1(c, e)),A^2 -> H(a, U1(c, e))) (A^2 -> H(a, U1(c, k)),A^2 -> H(a, U1(c, k))) ---------------------------------------- (1108) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, f(e)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1109) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(e)),A^2 -> H(c, f(e))) (A^2 -> H(d, f(e)),A^2 -> H(d, f(e))) (A^2 -> H(a, U1(e, e)),A^2 -> H(a, U1(e, e))) ---------------------------------------- (1110) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, f(k)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1111) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, f(k)),A^2 -> H(c, f(k))) (A^2 -> H(d, f(k)),A^2 -> H(d, f(k))) (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) ---------------------------------------- (1112) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(a, U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1113) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, d)),A^2 -> H(c, U1(d, d))) (A^2 -> H(d, U1(d, d)),A^2 -> H(d, U1(d, d))) (A^2 -> H(a, U1(k, d)),A^2 -> H(a, U1(k, d))) (A^2 -> H(a, U1(d, k)),A^2 -> H(a, U1(d, k))) ---------------------------------------- (1114) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(c, b)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1115) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(c, b)),A^2 -> H(d, U1(c, b))) (A^2 -> H(U1(e, k), U1(c, b)),A^2 -> H(U1(e, k), U1(c, b))) (A^2 -> H(U1(e, d), U1(e, b)),A^2 -> H(U1(e, d), U1(e, b))) (A^2 -> H(U1(e, d), U1(k, b)),A^2 -> H(U1(e, d), U1(k, b))) (A^2 -> H(U1(e, d), U1(c, c)),A^2 -> H(U1(e, d), U1(c, c))) (A^2 -> H(U1(e, d), U1(c, d)),A^2 -> H(U1(e, d), U1(c, d))) ---------------------------------------- (1116) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(d, b)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1117) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(d, b)),A^2 -> H(d, U1(d, b))) (A^2 -> H(U1(e, k), U1(d, b)),A^2 -> H(U1(e, k), U1(d, b))) (A^2 -> H(U1(e, d), U1(k, b)),A^2 -> H(U1(e, d), U1(k, b))) (A^2 -> H(U1(e, d), U1(d, c)),A^2 -> H(U1(e, d), U1(d, c))) (A^2 -> H(U1(e, d), U1(d, d)),A^2 -> H(U1(e, d), U1(d, d))) ---------------------------------------- (1118) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(b, c)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1119) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(b, c)),A^2 -> H(d, U1(b, c))) (A^2 -> H(U1(e, k), U1(b, c)),A^2 -> H(U1(e, k), U1(b, c))) (A^2 -> H(U1(e, d), U1(c, c)),A^2 -> H(U1(e, d), U1(c, c))) (A^2 -> H(U1(e, d), U1(d, c)),A^2 -> H(U1(e, d), U1(d, c))) (A^2 -> H(U1(e, d), U1(b, e)),A^2 -> H(U1(e, d), U1(b, e))) (A^2 -> H(U1(e, d), U1(b, k)),A^2 -> H(U1(e, d), U1(b, k))) ---------------------------------------- (1120) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(b, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1121) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(b, d)),A^2 -> H(d, U1(b, d))) (A^2 -> H(U1(e, k), U1(b, d)),A^2 -> H(U1(e, k), U1(b, d))) (A^2 -> H(U1(e, d), U1(c, d)),A^2 -> H(U1(e, d), U1(c, d))) (A^2 -> H(U1(e, d), U1(d, d)),A^2 -> H(U1(e, d), U1(d, d))) (A^2 -> H(U1(e, d), U1(b, k)),A^2 -> H(U1(e, d), U1(b, k))) ---------------------------------------- (1122) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(c, c)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1123) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(c, c)),A^2 -> H(d, U1(c, c))) (A^2 -> H(U1(e, k), U1(c, c)),A^2 -> H(U1(e, k), U1(c, c))) (A^2 -> H(U1(e, d), U1(e, c)),A^2 -> H(U1(e, d), U1(e, c))) (A^2 -> H(U1(e, d), U1(k, c)),A^2 -> H(U1(e, d), U1(k, c))) (A^2 -> H(U1(e, d), U1(c, e)),A^2 -> H(U1(e, d), U1(c, e))) (A^2 -> H(U1(e, d), U1(c, k)),A^2 -> H(U1(e, d), U1(c, k))) ---------------------------------------- (1124) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), f(e)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1125) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, f(e)),A^2 -> H(d, f(e))) (A^2 -> H(U1(e, k), f(e)),A^2 -> H(U1(e, k), f(e))) (A^2 -> H(U1(e, d), U1(e, e)),A^2 -> H(U1(e, d), U1(e, e))) ---------------------------------------- (1126) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), f(k)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1127) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, f(k)),A^2 -> H(d, f(k))) (A^2 -> H(U1(e, k), f(k)),A^2 -> H(U1(e, k), f(k))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1128) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1129) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(d, d)),A^2 -> H(d, U1(d, d))) (A^2 -> H(U1(e, k), U1(d, d)),A^2 -> H(U1(e, k), U1(d, d))) (A^2 -> H(U1(e, d), U1(k, d)),A^2 -> H(U1(e, d), U1(k, d))) (A^2 -> H(U1(e, d), U1(d, k)),A^2 -> H(U1(e, d), U1(d, k))) ---------------------------------------- (1130) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, b)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1131) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(e, b)),A^2 -> H(a, U1(e, b))) (A^2 -> H(U1(e, c), U1(e, b)),A^2 -> H(U1(e, c), U1(e, b))) (A^2 -> H(U1(e, d), U1(e, b)),A^2 -> H(U1(e, d), U1(e, b))) (A^2 -> H(U1(e, a), b),A^2 -> H(U1(e, a), b)) (A^2 -> H(U1(e, a), U1(e, c)),A^2 -> H(U1(e, a), U1(e, c))) (A^2 -> H(U1(e, a), U1(e, d)),A^2 -> H(U1(e, a), U1(e, d))) ---------------------------------------- (1132) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(k, b)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1133) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(k, b)),A^2 -> H(a, U1(k, b))) (A^2 -> H(U1(e, c), U1(k, b)),A^2 -> H(U1(e, c), U1(k, b))) (A^2 -> H(U1(e, d), U1(k, b)),A^2 -> H(U1(e, d), U1(k, b))) (A^2 -> H(U1(e, a), U1(k, c)),A^2 -> H(U1(e, a), U1(k, c))) (A^2 -> H(U1(e, a), U1(k, d)),A^2 -> H(U1(e, a), U1(k, d))) ---------------------------------------- (1134) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(c, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1135) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(c, d)),A^2 -> H(a, U1(c, d))) (A^2 -> H(U1(e, c), U1(c, d)),A^2 -> H(U1(e, c), U1(c, d))) (A^2 -> H(U1(e, d), U1(c, d)),A^2 -> H(U1(e, d), U1(c, d))) (A^2 -> H(U1(e, a), U1(e, d)),A^2 -> H(U1(e, a), U1(e, d))) (A^2 -> H(U1(e, a), U1(k, d)),A^2 -> H(U1(e, a), U1(k, d))) (A^2 -> H(U1(e, a), U1(c, k)),A^2 -> H(U1(e, a), U1(c, k))) ---------------------------------------- (1136) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(d, c)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1137) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(d, c)),A^2 -> H(a, U1(d, c))) (A^2 -> H(U1(e, c), U1(d, c)),A^2 -> H(U1(e, c), U1(d, c))) (A^2 -> H(U1(e, d), U1(d, c)),A^2 -> H(U1(e, d), U1(d, c))) (A^2 -> H(U1(e, a), U1(k, c)),A^2 -> H(U1(e, a), U1(k, c))) (A^2 -> H(U1(e, a), U1(d, e)),A^2 -> H(U1(e, a), U1(d, e))) (A^2 -> H(U1(e, a), U1(d, k)),A^2 -> H(U1(e, a), U1(d, k))) ---------------------------------------- (1138) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(b, e)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1139) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(b, e)),A^2 -> H(a, U1(b, e))) (A^2 -> H(U1(e, c), U1(b, e)),A^2 -> H(U1(e, c), U1(b, e))) (A^2 -> H(U1(e, d), U1(b, e)),A^2 -> H(U1(e, d), U1(b, e))) (A^2 -> H(U1(e, a), U1(c, e)),A^2 -> H(U1(e, a), U1(c, e))) (A^2 -> H(U1(e, a), U1(d, e)),A^2 -> H(U1(e, a), U1(d, e))) ---------------------------------------- (1140) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(b, k)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1141) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(b, k)),A^2 -> H(a, U1(b, k))) (A^2 -> H(U1(e, c), U1(b, k)),A^2 -> H(U1(e, c), U1(b, k))) (A^2 -> H(U1(e, d), U1(b, k)),A^2 -> H(U1(e, d), U1(b, k))) (A^2 -> H(U1(e, a), U1(c, k)),A^2 -> H(U1(e, a), U1(c, k))) (A^2 -> H(U1(e, a), U1(d, k)),A^2 -> H(U1(e, a), U1(d, k))) ---------------------------------------- (1142) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, c)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1143) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(e, c)),A^2 -> H(a, U1(e, c))) (A^2 -> H(U1(e, c), U1(e, c)),A^2 -> H(U1(e, c), U1(e, c))) (A^2 -> H(U1(e, d), U1(e, c)),A^2 -> H(U1(e, d), U1(e, c))) (A^2 -> H(U1(e, a), c),A^2 -> H(U1(e, a), c)) (A^2 -> H(U1(e, a), U1(e, e)),A^2 -> H(U1(e, a), U1(e, e))) (A^2 -> H(U1(e, a), U1(e, k)),A^2 -> H(U1(e, a), U1(e, k))) ---------------------------------------- (1144) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(k, c)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1145) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(k, c)),A^2 -> H(a, U1(k, c))) (A^2 -> H(U1(e, c), U1(k, c)),A^2 -> H(U1(e, c), U1(k, c))) (A^2 -> H(U1(e, d), U1(k, c)),A^2 -> H(U1(e, d), U1(k, c))) (A^2 -> H(U1(e, a), U1(k, e)),A^2 -> H(U1(e, a), U1(k, e))) (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) ---------------------------------------- (1146) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(c, e)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1147) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(c, e)),A^2 -> H(a, U1(c, e))) (A^2 -> H(U1(e, c), U1(c, e)),A^2 -> H(U1(e, c), U1(c, e))) (A^2 -> H(U1(e, d), U1(c, e)),A^2 -> H(U1(e, d), U1(c, e))) (A^2 -> H(U1(e, a), U1(e, e)),A^2 -> H(U1(e, a), U1(e, e))) (A^2 -> H(U1(e, a), U1(k, e)),A^2 -> H(U1(e, a), U1(k, e))) ---------------------------------------- (1148) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(c, k)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1149) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(c, k)),A^2 -> H(a, U1(c, k))) (A^2 -> H(U1(e, c), U1(c, k)),A^2 -> H(U1(e, c), U1(c, k))) (A^2 -> H(U1(e, d), U1(c, k)),A^2 -> H(U1(e, d), U1(c, k))) (A^2 -> H(U1(e, a), U1(e, k)),A^2 -> H(U1(e, a), U1(e, k))) (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) ---------------------------------------- (1150) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(e, e)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1151) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(e, e)),A^2 -> H(a, U1(e, e))) (A^2 -> H(U1(e, c), U1(e, e)),A^2 -> H(U1(e, c), U1(e, e))) (A^2 -> H(U1(e, d), U1(e, e)),A^2 -> H(U1(e, d), U1(e, e))) (A^2 -> H(U1(e, a), e),A^2 -> H(U1(e, a), e)) ---------------------------------------- (1152) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1153) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1154) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(k, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1155) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(k, d)),A^2 -> H(a, U1(k, d))) (A^2 -> H(U1(e, c), U1(k, d)),A^2 -> H(U1(e, c), U1(k, d))) (A^2 -> H(U1(e, d), U1(k, d)),A^2 -> H(U1(e, d), U1(k, d))) (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) ---------------------------------------- (1156) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(e, a), U1(d, k)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1157) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(d, k)),A^2 -> H(a, U1(d, k))) (A^2 -> H(U1(e, c), U1(d, k)),A^2 -> H(U1(e, c), U1(d, k))) (A^2 -> H(U1(e, d), U1(d, k)),A^2 -> H(U1(e, d), U1(d, k))) (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) ---------------------------------------- (1158) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(e, b)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1159) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, b)),A^2 -> H(U1(k, c), U1(e, b))) (A^2 -> H(U1(k, d), U1(e, b)),A^2 -> H(U1(k, d), U1(e, b))) (A^2 -> H(U1(k, a), b),A^2 -> H(U1(k, a), b)) (A^2 -> H(U1(k, a), U1(e, c)),A^2 -> H(U1(k, a), U1(e, c))) (A^2 -> H(U1(k, a), U1(e, d)),A^2 -> H(U1(k, a), U1(e, d))) ---------------------------------------- (1160) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(k, b)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1161) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, b)),A^2 -> H(U1(k, c), U1(k, b))) (A^2 -> H(U1(k, d), U1(k, b)),A^2 -> H(U1(k, d), U1(k, b))) (A^2 -> H(U1(k, a), U1(k, c)),A^2 -> H(U1(k, a), U1(k, c))) (A^2 -> H(U1(k, a), U1(k, d)),A^2 -> H(U1(k, a), U1(k, d))) ---------------------------------------- (1162) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(c, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1163) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, d)),A^2 -> H(U1(k, c), U1(c, d))) (A^2 -> H(U1(k, d), U1(c, d)),A^2 -> H(U1(k, d), U1(c, d))) (A^2 -> H(U1(k, a), U1(e, d)),A^2 -> H(U1(k, a), U1(e, d))) (A^2 -> H(U1(k, a), U1(k, d)),A^2 -> H(U1(k, a), U1(k, d))) (A^2 -> H(U1(k, a), U1(c, k)),A^2 -> H(U1(k, a), U1(c, k))) ---------------------------------------- (1164) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(d, c)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1165) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, c)),A^2 -> H(U1(k, c), U1(d, c))) (A^2 -> H(U1(k, d), U1(d, c)),A^2 -> H(U1(k, d), U1(d, c))) (A^2 -> H(U1(k, a), U1(k, c)),A^2 -> H(U1(k, a), U1(k, c))) (A^2 -> H(U1(k, a), U1(d, e)),A^2 -> H(U1(k, a), U1(d, e))) (A^2 -> H(U1(k, a), U1(d, k)),A^2 -> H(U1(k, a), U1(d, k))) ---------------------------------------- (1166) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(b, e)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1167) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, e)),A^2 -> H(U1(k, c), U1(b, e))) (A^2 -> H(U1(k, d), U1(b, e)),A^2 -> H(U1(k, d), U1(b, e))) (A^2 -> H(U1(k, a), U1(c, e)),A^2 -> H(U1(k, a), U1(c, e))) (A^2 -> H(U1(k, a), U1(d, e)),A^2 -> H(U1(k, a), U1(d, e))) ---------------------------------------- (1168) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(b, k)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1169) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, k)),A^2 -> H(U1(k, c), U1(b, k))) (A^2 -> H(U1(k, d), U1(b, k)),A^2 -> H(U1(k, d), U1(b, k))) (A^2 -> H(U1(k, a), U1(c, k)),A^2 -> H(U1(k, a), U1(c, k))) (A^2 -> H(U1(k, a), U1(d, k)),A^2 -> H(U1(k, a), U1(d, k))) ---------------------------------------- (1170) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(e, c)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1171) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, c)),A^2 -> H(U1(k, c), U1(e, c))) (A^2 -> H(U1(k, d), U1(e, c)),A^2 -> H(U1(k, d), U1(e, c))) (A^2 -> H(U1(k, a), c),A^2 -> H(U1(k, a), c)) (A^2 -> H(U1(k, a), U1(e, e)),A^2 -> H(U1(k, a), U1(e, e))) (A^2 -> H(U1(k, a), U1(e, k)),A^2 -> H(U1(k, a), U1(e, k))) ---------------------------------------- (1172) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(k, c)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1173) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, c)),A^2 -> H(U1(k, c), U1(k, c))) (A^2 -> H(U1(k, d), U1(k, c)),A^2 -> H(U1(k, d), U1(k, c))) (A^2 -> H(U1(k, a), U1(k, e)),A^2 -> H(U1(k, a), U1(k, e))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) ---------------------------------------- (1174) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(c, e)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1175) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, e)),A^2 -> H(U1(k, c), U1(c, e))) (A^2 -> H(U1(k, d), U1(c, e)),A^2 -> H(U1(k, d), U1(c, e))) (A^2 -> H(U1(k, a), U1(e, e)),A^2 -> H(U1(k, a), U1(e, e))) (A^2 -> H(U1(k, a), U1(k, e)),A^2 -> H(U1(k, a), U1(k, e))) ---------------------------------------- (1176) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(c, k)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1177) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, k)),A^2 -> H(U1(k, c), U1(c, k))) (A^2 -> H(U1(k, d), U1(c, k)),A^2 -> H(U1(k, d), U1(c, k))) (A^2 -> H(U1(k, a), U1(e, k)),A^2 -> H(U1(k, a), U1(e, k))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) ---------------------------------------- (1178) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(e, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1179) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) (A^2 -> H(U1(k, d), U1(e, e)),A^2 -> H(U1(k, d), U1(e, e))) (A^2 -> H(U1(k, a), e),A^2 -> H(U1(k, a), e)) ---------------------------------------- (1180) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1181) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1182) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1183) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) ---------------------------------------- (1184) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(k, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1185) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, d)),A^2 -> H(U1(k, c), U1(k, d))) (A^2 -> H(U1(k, d), U1(k, d)),A^2 -> H(U1(k, d), U1(k, d))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) ---------------------------------------- (1186) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(k, a), U1(d, k)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1187) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, k)),A^2 -> H(U1(k, c), U1(d, k))) (A^2 -> H(U1(k, d), U1(d, k)),A^2 -> H(U1(k, d), U1(d, k))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) ---------------------------------------- (1188) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(e, b)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1189) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(e, b)),A^2 -> H(U1(e, d), U1(e, b))) (A^2 -> H(U1(k, d), U1(e, b)),A^2 -> H(U1(k, d), U1(e, b))) (A^2 -> H(U1(c, k), U1(e, b)),A^2 -> H(U1(c, k), U1(e, b))) (A^2 -> H(U1(c, d), b),A^2 -> H(U1(c, d), b)) (A^2 -> H(U1(c, d), U1(e, c)),A^2 -> H(U1(c, d), U1(e, c))) (A^2 -> H(U1(c, d), U1(e, d)),A^2 -> H(U1(c, d), U1(e, d))) ---------------------------------------- (1190) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(k, b)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1191) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(k, b)),A^2 -> H(U1(e, d), U1(k, b))) (A^2 -> H(U1(k, d), U1(k, b)),A^2 -> H(U1(k, d), U1(k, b))) (A^2 -> H(U1(c, k), U1(k, b)),A^2 -> H(U1(c, k), U1(k, b))) (A^2 -> H(U1(c, d), U1(k, c)),A^2 -> H(U1(c, d), U1(k, c))) (A^2 -> H(U1(c, d), U1(k, d)),A^2 -> H(U1(c, d), U1(k, d))) ---------------------------------------- (1192) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, c)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1193) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(d, c)),A^2 -> H(U1(e, d), U1(d, c))) (A^2 -> H(U1(k, d), U1(d, c)),A^2 -> H(U1(k, d), U1(d, c))) (A^2 -> H(U1(c, k), U1(d, c)),A^2 -> H(U1(c, k), U1(d, c))) (A^2 -> H(U1(c, d), U1(k, c)),A^2 -> H(U1(c, d), U1(k, c))) (A^2 -> H(U1(c, d), U1(d, e)),A^2 -> H(U1(c, d), U1(d, e))) (A^2 -> H(U1(c, d), U1(d, k)),A^2 -> H(U1(c, d), U1(d, k))) ---------------------------------------- (1194) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(b, e)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1195) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(b, e)),A^2 -> H(U1(e, d), U1(b, e))) (A^2 -> H(U1(k, d), U1(b, e)),A^2 -> H(U1(k, d), U1(b, e))) (A^2 -> H(U1(c, k), U1(b, e)),A^2 -> H(U1(c, k), U1(b, e))) (A^2 -> H(U1(c, d), U1(c, e)),A^2 -> H(U1(c, d), U1(c, e))) (A^2 -> H(U1(c, d), U1(d, e)),A^2 -> H(U1(c, d), U1(d, e))) ---------------------------------------- (1196) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(b, k)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1197) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(b, k)),A^2 -> H(U1(e, d), U1(b, k))) (A^2 -> H(U1(k, d), U1(b, k)),A^2 -> H(U1(k, d), U1(b, k))) (A^2 -> H(U1(c, k), U1(b, k)),A^2 -> H(U1(c, k), U1(b, k))) (A^2 -> H(U1(c, d), U1(c, k)),A^2 -> H(U1(c, d), U1(c, k))) (A^2 -> H(U1(c, d), U1(d, k)),A^2 -> H(U1(c, d), U1(d, k))) ---------------------------------------- (1198) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(e, c)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1199) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(e, c)),A^2 -> H(U1(e, d), U1(e, c))) (A^2 -> H(U1(k, d), U1(e, c)),A^2 -> H(U1(k, d), U1(e, c))) (A^2 -> H(U1(c, k), U1(e, c)),A^2 -> H(U1(c, k), U1(e, c))) (A^2 -> H(U1(c, d), c),A^2 -> H(U1(c, d), c)) (A^2 -> H(U1(c, d), U1(e, e)),A^2 -> H(U1(c, d), U1(e, e))) (A^2 -> H(U1(c, d), U1(e, k)),A^2 -> H(U1(c, d), U1(e, k))) ---------------------------------------- (1200) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(k, c)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1201) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(k, c)),A^2 -> H(U1(e, d), U1(k, c))) (A^2 -> H(U1(k, d), U1(k, c)),A^2 -> H(U1(k, d), U1(k, c))) (A^2 -> H(U1(c, k), U1(k, c)),A^2 -> H(U1(c, k), U1(k, c))) (A^2 -> H(U1(c, d), U1(k, e)),A^2 -> H(U1(c, d), U1(k, e))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (1202) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(c, e)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1203) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(c, e)),A^2 -> H(U1(e, d), U1(c, e))) (A^2 -> H(U1(k, d), U1(c, e)),A^2 -> H(U1(k, d), U1(c, e))) (A^2 -> H(U1(c, k), U1(c, e)),A^2 -> H(U1(c, k), U1(c, e))) (A^2 -> H(U1(c, d), U1(e, e)),A^2 -> H(U1(c, d), U1(e, e))) (A^2 -> H(U1(c, d), U1(k, e)),A^2 -> H(U1(c, d), U1(k, e))) ---------------------------------------- (1204) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(c, k)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1205) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(c, k)),A^2 -> H(U1(e, d), U1(c, k))) (A^2 -> H(U1(k, d), U1(c, k)),A^2 -> H(U1(k, d), U1(c, k))) (A^2 -> H(U1(c, k), U1(c, k)),A^2 -> H(U1(c, k), U1(c, k))) (A^2 -> H(U1(c, d), U1(e, k)),A^2 -> H(U1(c, d), U1(e, k))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (1206) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(e, e)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1207) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(e, e)),A^2 -> H(U1(e, d), U1(e, e))) (A^2 -> H(U1(k, d), U1(e, e)),A^2 -> H(U1(k, d), U1(e, e))) (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) (A^2 -> H(U1(c, d), e),A^2 -> H(U1(c, d), e)) ---------------------------------------- (1208) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1209) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (1210) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(k, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1211) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(k, d)),A^2 -> H(U1(e, d), U1(k, d))) (A^2 -> H(U1(k, d), U1(k, d)),A^2 -> H(U1(k, d), U1(k, d))) (A^2 -> H(U1(c, k), U1(k, d)),A^2 -> H(U1(c, k), U1(k, d))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (1212) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, d), U1(d, k)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1213) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(d, k)),A^2 -> H(U1(e, d), U1(d, k))) (A^2 -> H(U1(k, d), U1(d, k)),A^2 -> H(U1(k, d), U1(d, k))) (A^2 -> H(U1(c, k), U1(d, k)),A^2 -> H(U1(c, k), U1(d, k))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (1214) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), b) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1215) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), b),A^2 -> H(U1(e, a), b)) (A^2 -> H(U1(k, a), b),A^2 -> H(U1(k, a), b)) (A^2 -> H(U1(c, c), b),A^2 -> H(U1(c, c), b)) (A^2 -> H(U1(c, d), b),A^2 -> H(U1(c, d), b)) (A^2 -> H(U1(c, a), c),A^2 -> H(U1(c, a), c)) (A^2 -> H(U1(c, a), d),A^2 -> H(U1(c, a), d)) ---------------------------------------- (1216) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), U1(e, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1217) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(e, d)),A^2 -> H(U1(e, a), U1(e, d))) (A^2 -> H(U1(k, a), U1(e, d)),A^2 -> H(U1(k, a), U1(e, d))) (A^2 -> H(U1(c, c), U1(e, d)),A^2 -> H(U1(c, c), U1(e, d))) (A^2 -> H(U1(c, d), U1(e, d)),A^2 -> H(U1(c, d), U1(e, d))) (A^2 -> H(U1(c, a), d),A^2 -> H(U1(c, a), d)) (A^2 -> H(U1(c, a), U1(e, k)),A^2 -> H(U1(c, a), U1(e, k))) ---------------------------------------- (1218) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), U1(d, e)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1219) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(d, e)),A^2 -> H(U1(e, a), U1(d, e))) (A^2 -> H(U1(k, a), U1(d, e)),A^2 -> H(U1(k, a), U1(d, e))) (A^2 -> H(U1(c, c), U1(d, e)),A^2 -> H(U1(c, c), U1(d, e))) (A^2 -> H(U1(c, d), U1(d, e)),A^2 -> H(U1(c, d), U1(d, e))) (A^2 -> H(U1(c, a), U1(k, e)),A^2 -> H(U1(c, a), U1(k, e))) ---------------------------------------- (1220) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), c) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1221) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), c),A^2 -> H(U1(e, a), c)) (A^2 -> H(U1(k, a), c),A^2 -> H(U1(k, a), c)) (A^2 -> H(U1(c, c), c),A^2 -> H(U1(c, c), c)) (A^2 -> H(U1(c, d), c),A^2 -> H(U1(c, d), c)) (A^2 -> H(U1(c, a), e),A^2 -> H(U1(c, a), e)) (A^2 -> H(U1(c, a), k),A^2 -> H(U1(c, a), k)) ---------------------------------------- (1222) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), U1(e, k)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1223) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(e, k)),A^2 -> H(U1(e, a), U1(e, k))) (A^2 -> H(U1(k, a), U1(e, k)),A^2 -> H(U1(k, a), U1(e, k))) (A^2 -> H(U1(c, c), U1(e, k)),A^2 -> H(U1(c, c), U1(e, k))) (A^2 -> H(U1(c, d), U1(e, k)),A^2 -> H(U1(c, d), U1(e, k))) (A^2 -> H(U1(c, a), k),A^2 -> H(U1(c, a), k)) ---------------------------------------- (1224) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), U1(k, e)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1225) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(k, e)),A^2 -> H(U1(e, a), U1(k, e))) (A^2 -> H(U1(k, a), U1(k, e)),A^2 -> H(U1(k, a), U1(k, e))) (A^2 -> H(U1(c, c), U1(k, e)),A^2 -> H(U1(c, c), U1(k, e))) (A^2 -> H(U1(c, d), U1(k, e)),A^2 -> H(U1(c, d), U1(k, e))) ---------------------------------------- (1226) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), e) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1227) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), e),A^2 -> H(U1(e, a), e)) (A^2 -> H(U1(k, a), e),A^2 -> H(U1(k, a), e)) (A^2 -> H(U1(c, c), e),A^2 -> H(U1(c, c), e)) (A^2 -> H(U1(c, d), e),A^2 -> H(U1(c, d), e)) ---------------------------------------- (1228) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(k, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1229) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1230) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(c, a), U1(k, k)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1231) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), U1(k, k)),A^2 -> H(U1(e, a), U1(k, k))) (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) ---------------------------------------- (1232) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), U1(c, b)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1233) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, b)),A^2 -> H(U1(k, e), U1(c, b))) (A^2 -> H(U1(d, e), U1(e, b)),A^2 -> H(U1(d, e), U1(e, b))) (A^2 -> H(U1(d, e), U1(k, b)),A^2 -> H(U1(d, e), U1(k, b))) (A^2 -> H(U1(d, e), U1(c, c)),A^2 -> H(U1(d, e), U1(c, c))) (A^2 -> H(U1(d, e), U1(c, d)),A^2 -> H(U1(d, e), U1(c, d))) ---------------------------------------- (1234) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), U1(d, b)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1235) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, b)),A^2 -> H(U1(k, e), U1(d, b))) (A^2 -> H(U1(d, e), U1(k, b)),A^2 -> H(U1(d, e), U1(k, b))) (A^2 -> H(U1(d, e), U1(d, c)),A^2 -> H(U1(d, e), U1(d, c))) (A^2 -> H(U1(d, e), U1(d, d)),A^2 -> H(U1(d, e), U1(d, d))) ---------------------------------------- (1236) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), U1(b, c)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1237) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, c)),A^2 -> H(U1(k, e), U1(b, c))) (A^2 -> H(U1(d, e), U1(c, c)),A^2 -> H(U1(d, e), U1(c, c))) (A^2 -> H(U1(d, e), U1(d, c)),A^2 -> H(U1(d, e), U1(d, c))) (A^2 -> H(U1(d, e), U1(b, e)),A^2 -> H(U1(d, e), U1(b, e))) (A^2 -> H(U1(d, e), U1(b, k)),A^2 -> H(U1(d, e), U1(b, k))) ---------------------------------------- (1238) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), U1(b, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1239) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, d)),A^2 -> H(U1(k, e), U1(b, d))) (A^2 -> H(U1(d, e), U1(c, d)),A^2 -> H(U1(d, e), U1(c, d))) (A^2 -> H(U1(d, e), U1(d, d)),A^2 -> H(U1(d, e), U1(d, d))) (A^2 -> H(U1(d, e), U1(b, k)),A^2 -> H(U1(d, e), U1(b, k))) ---------------------------------------- (1240) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), U1(c, c)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1241) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, c)),A^2 -> H(U1(k, e), U1(c, c))) (A^2 -> H(U1(d, e), U1(e, c)),A^2 -> H(U1(d, e), U1(e, c))) (A^2 -> H(U1(d, e), U1(k, c)),A^2 -> H(U1(d, e), U1(k, c))) (A^2 -> H(U1(d, e), U1(c, e)),A^2 -> H(U1(d, e), U1(c, e))) (A^2 -> H(U1(d, e), U1(c, k)),A^2 -> H(U1(d, e), U1(c, k))) ---------------------------------------- (1242) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), f(e)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1243) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(e)),A^2 -> H(U1(k, e), f(e))) (A^2 -> H(U1(d, e), U1(e, e)),A^2 -> H(U1(d, e), U1(e, e))) ---------------------------------------- (1244) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), f(k)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1245) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), f(k)),A^2 -> H(U1(k, e), f(k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1246) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, e), U1(d, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1247) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, d)),A^2 -> H(U1(k, e), U1(d, d))) (A^2 -> H(U1(d, e), U1(k, d)),A^2 -> H(U1(d, e), U1(k, d))) (A^2 -> H(U1(d, e), U1(d, k)),A^2 -> H(U1(d, e), U1(d, k))) ---------------------------------------- (1248) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(e, b)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1249) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, b)),A^2 -> H(U1(k, c), U1(e, b))) (A^2 -> H(U1(d, e), U1(e, b)),A^2 -> H(U1(d, e), U1(e, b))) (A^2 -> H(U1(d, k), U1(e, b)),A^2 -> H(U1(d, k), U1(e, b))) (A^2 -> H(U1(d, c), b),A^2 -> H(U1(d, c), b)) (A^2 -> H(U1(d, c), U1(e, c)),A^2 -> H(U1(d, c), U1(e, c))) (A^2 -> H(U1(d, c), U1(e, d)),A^2 -> H(U1(d, c), U1(e, d))) ---------------------------------------- (1250) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(k, b)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1251) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, b)),A^2 -> H(U1(k, c), U1(k, b))) (A^2 -> H(U1(d, e), U1(k, b)),A^2 -> H(U1(d, e), U1(k, b))) (A^2 -> H(U1(d, k), U1(k, b)),A^2 -> H(U1(d, k), U1(k, b))) (A^2 -> H(U1(d, c), U1(k, c)),A^2 -> H(U1(d, c), U1(k, c))) (A^2 -> H(U1(d, c), U1(k, d)),A^2 -> H(U1(d, c), U1(k, d))) ---------------------------------------- (1252) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1253) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, d)),A^2 -> H(U1(k, c), U1(c, d))) (A^2 -> H(U1(d, e), U1(c, d)),A^2 -> H(U1(d, e), U1(c, d))) (A^2 -> H(U1(d, k), U1(c, d)),A^2 -> H(U1(d, k), U1(c, d))) (A^2 -> H(U1(d, c), U1(e, d)),A^2 -> H(U1(d, c), U1(e, d))) (A^2 -> H(U1(d, c), U1(k, d)),A^2 -> H(U1(d, c), U1(k, d))) (A^2 -> H(U1(d, c), U1(c, k)),A^2 -> H(U1(d, c), U1(c, k))) ---------------------------------------- (1254) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, e)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1255) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, e)),A^2 -> H(U1(k, c), U1(b, e))) (A^2 -> H(U1(d, e), U1(b, e)),A^2 -> H(U1(d, e), U1(b, e))) (A^2 -> H(U1(d, k), U1(b, e)),A^2 -> H(U1(d, k), U1(b, e))) (A^2 -> H(U1(d, c), U1(c, e)),A^2 -> H(U1(d, c), U1(c, e))) (A^2 -> H(U1(d, c), U1(d, e)),A^2 -> H(U1(d, c), U1(d, e))) ---------------------------------------- (1256) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(b, k)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1257) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(b, k)),A^2 -> H(U1(k, c), U1(b, k))) (A^2 -> H(U1(d, e), U1(b, k)),A^2 -> H(U1(d, e), U1(b, k))) (A^2 -> H(U1(d, k), U1(b, k)),A^2 -> H(U1(d, k), U1(b, k))) (A^2 -> H(U1(d, c), U1(c, k)),A^2 -> H(U1(d, c), U1(c, k))) (A^2 -> H(U1(d, c), U1(d, k)),A^2 -> H(U1(d, c), U1(d, k))) ---------------------------------------- (1258) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(e, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1259) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, c)),A^2 -> H(U1(k, c), U1(e, c))) (A^2 -> H(U1(d, e), U1(e, c)),A^2 -> H(U1(d, e), U1(e, c))) (A^2 -> H(U1(d, k), U1(e, c)),A^2 -> H(U1(d, k), U1(e, c))) (A^2 -> H(U1(d, c), c),A^2 -> H(U1(d, c), c)) (A^2 -> H(U1(d, c), U1(e, e)),A^2 -> H(U1(d, c), U1(e, e))) (A^2 -> H(U1(d, c), U1(e, k)),A^2 -> H(U1(d, c), U1(e, k))) ---------------------------------------- (1260) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(k, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1261) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, c)),A^2 -> H(U1(k, c), U1(k, c))) (A^2 -> H(U1(d, e), U1(k, c)),A^2 -> H(U1(d, e), U1(k, c))) (A^2 -> H(U1(d, k), U1(k, c)),A^2 -> H(U1(d, k), U1(k, c))) (A^2 -> H(U1(d, c), U1(k, e)),A^2 -> H(U1(d, c), U1(k, e))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) ---------------------------------------- (1262) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(c, e)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1263) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, e)),A^2 -> H(U1(k, c), U1(c, e))) (A^2 -> H(U1(d, e), U1(c, e)),A^2 -> H(U1(d, e), U1(c, e))) (A^2 -> H(U1(d, k), U1(c, e)),A^2 -> H(U1(d, k), U1(c, e))) (A^2 -> H(U1(d, c), U1(e, e)),A^2 -> H(U1(d, c), U1(e, e))) (A^2 -> H(U1(d, c), U1(k, e)),A^2 -> H(U1(d, c), U1(k, e))) ---------------------------------------- (1264) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(c, k)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1265) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(c, k)),A^2 -> H(U1(k, c), U1(c, k))) (A^2 -> H(U1(d, e), U1(c, k)),A^2 -> H(U1(d, e), U1(c, k))) (A^2 -> H(U1(d, k), U1(c, k)),A^2 -> H(U1(d, k), U1(c, k))) (A^2 -> H(U1(d, c), U1(e, k)),A^2 -> H(U1(d, c), U1(e, k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) ---------------------------------------- (1266) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(e, e)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1267) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) (A^2 -> H(U1(d, e), U1(e, e)),A^2 -> H(U1(d, e), U1(e, e))) (A^2 -> H(U1(d, k), U1(e, e)),A^2 -> H(U1(d, k), U1(e, e))) (A^2 -> H(U1(d, c), e),A^2 -> H(U1(d, c), e)) ---------------------------------------- (1268) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1269) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1270) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(k, d)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1271) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, d)),A^2 -> H(U1(k, c), U1(k, d))) (A^2 -> H(U1(d, e), U1(k, d)),A^2 -> H(U1(d, e), U1(k, d))) (A^2 -> H(U1(d, k), U1(k, d)),A^2 -> H(U1(d, k), U1(k, d))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) ---------------------------------------- (1272) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, c), U1(d, k)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1273) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, k)),A^2 -> H(U1(k, c), U1(d, k))) (A^2 -> H(U1(d, e), U1(d, k)),A^2 -> H(U1(d, e), U1(d, k))) (A^2 -> H(U1(d, k), U1(d, k)),A^2 -> H(U1(d, k), U1(d, k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) ---------------------------------------- (1274) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), b) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1275) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), b),A^2 -> H(U1(k, a), b)) (A^2 -> H(U1(d, c), b),A^2 -> H(U1(d, c), b)) (A^2 -> H(U1(d, d), b),A^2 -> H(U1(d, d), b)) (A^2 -> H(U1(d, a), c),A^2 -> H(U1(d, a), c)) (A^2 -> H(U1(d, a), d),A^2 -> H(U1(d, a), d)) ---------------------------------------- (1276) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), U1(e, d)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1277) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(e, d)),A^2 -> H(U1(k, a), U1(e, d))) (A^2 -> H(U1(d, c), U1(e, d)),A^2 -> H(U1(d, c), U1(e, d))) (A^2 -> H(U1(d, d), U1(e, d)),A^2 -> H(U1(d, d), U1(e, d))) (A^2 -> H(U1(d, a), d),A^2 -> H(U1(d, a), d)) (A^2 -> H(U1(d, a), U1(e, k)),A^2 -> H(U1(d, a), U1(e, k))) ---------------------------------------- (1278) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), U1(d, e)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1279) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(d, e)),A^2 -> H(U1(k, a), U1(d, e))) (A^2 -> H(U1(d, c), U1(d, e)),A^2 -> H(U1(d, c), U1(d, e))) (A^2 -> H(U1(d, d), U1(d, e)),A^2 -> H(U1(d, d), U1(d, e))) (A^2 -> H(U1(d, a), U1(k, e)),A^2 -> H(U1(d, a), U1(k, e))) ---------------------------------------- (1280) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), c) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1281) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), c),A^2 -> H(U1(k, a), c)) (A^2 -> H(U1(d, c), c),A^2 -> H(U1(d, c), c)) (A^2 -> H(U1(d, d), c),A^2 -> H(U1(d, d), c)) (A^2 -> H(U1(d, a), e),A^2 -> H(U1(d, a), e)) (A^2 -> H(U1(d, a), k),A^2 -> H(U1(d, a), k)) ---------------------------------------- (1282) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), U1(e, k)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1283) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(e, k)),A^2 -> H(U1(k, a), U1(e, k))) (A^2 -> H(U1(d, c), U1(e, k)),A^2 -> H(U1(d, c), U1(e, k))) (A^2 -> H(U1(d, d), U1(e, k)),A^2 -> H(U1(d, d), U1(e, k))) (A^2 -> H(U1(d, a), k),A^2 -> H(U1(d, a), k)) ---------------------------------------- (1284) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), U1(k, e)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1285) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(k, e)),A^2 -> H(U1(k, a), U1(k, e))) (A^2 -> H(U1(d, c), U1(k, e)),A^2 -> H(U1(d, c), U1(k, e))) (A^2 -> H(U1(d, d), U1(k, e)),A^2 -> H(U1(d, d), U1(k, e))) ---------------------------------------- (1286) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), e) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1287) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), e),A^2 -> H(U1(k, a), e)) (A^2 -> H(U1(d, c), e),A^2 -> H(U1(d, c), e)) (A^2 -> H(U1(d, d), e),A^2 -> H(U1(d, d), e)) ---------------------------------------- (1288) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(k, a), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1289) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1290) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(d, a), U1(k, k)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1291) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), U1(k, k)),A^2 -> H(U1(k, a), U1(k, k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) ---------------------------------------- (1292) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(e, b)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1293) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(e, b)),A^2 -> H(U1(c, e), U1(e, b))) (A^2 -> H(U1(d, e), U1(e, b)),A^2 -> H(U1(d, e), U1(e, b))) (A^2 -> H(U1(a, e), b),A^2 -> H(U1(a, e), b)) (A^2 -> H(U1(a, e), U1(e, c)),A^2 -> H(U1(a, e), U1(e, c))) (A^2 -> H(U1(a, e), U1(e, d)),A^2 -> H(U1(a, e), U1(e, d))) ---------------------------------------- (1294) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(k, b)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1295) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(k, b)),A^2 -> H(U1(c, e), U1(k, b))) (A^2 -> H(U1(d, e), U1(k, b)),A^2 -> H(U1(d, e), U1(k, b))) (A^2 -> H(U1(a, e), U1(k, c)),A^2 -> H(U1(a, e), U1(k, c))) (A^2 -> H(U1(a, e), U1(k, d)),A^2 -> H(U1(a, e), U1(k, d))) ---------------------------------------- (1296) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(c, d)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1297) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(c, d)),A^2 -> H(U1(c, e), U1(c, d))) (A^2 -> H(U1(d, e), U1(c, d)),A^2 -> H(U1(d, e), U1(c, d))) (A^2 -> H(U1(a, e), U1(e, d)),A^2 -> H(U1(a, e), U1(e, d))) (A^2 -> H(U1(a, e), U1(k, d)),A^2 -> H(U1(a, e), U1(k, d))) (A^2 -> H(U1(a, e), U1(c, k)),A^2 -> H(U1(a, e), U1(c, k))) ---------------------------------------- (1298) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1299) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(d, c)),A^2 -> H(U1(c, e), U1(d, c))) (A^2 -> H(U1(d, e), U1(d, c)),A^2 -> H(U1(d, e), U1(d, c))) (A^2 -> H(U1(a, e), U1(k, c)),A^2 -> H(U1(a, e), U1(k, c))) (A^2 -> H(U1(a, e), U1(d, e)),A^2 -> H(U1(a, e), U1(d, e))) (A^2 -> H(U1(a, e), U1(d, k)),A^2 -> H(U1(a, e), U1(d, k))) ---------------------------------------- (1300) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(b, e)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1301) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(b, e)),A^2 -> H(U1(c, e), U1(b, e))) (A^2 -> H(U1(d, e), U1(b, e)),A^2 -> H(U1(d, e), U1(b, e))) (A^2 -> H(U1(a, e), U1(c, e)),A^2 -> H(U1(a, e), U1(c, e))) (A^2 -> H(U1(a, e), U1(d, e)),A^2 -> H(U1(a, e), U1(d, e))) ---------------------------------------- (1302) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(b, k)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1303) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(b, k)),A^2 -> H(U1(c, e), U1(b, k))) (A^2 -> H(U1(d, e), U1(b, k)),A^2 -> H(U1(d, e), U1(b, k))) (A^2 -> H(U1(a, e), U1(c, k)),A^2 -> H(U1(a, e), U1(c, k))) (A^2 -> H(U1(a, e), U1(d, k)),A^2 -> H(U1(a, e), U1(d, k))) ---------------------------------------- (1304) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(e, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1305) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(e, c)),A^2 -> H(U1(c, e), U1(e, c))) (A^2 -> H(U1(d, e), U1(e, c)),A^2 -> H(U1(d, e), U1(e, c))) (A^2 -> H(U1(a, e), c),A^2 -> H(U1(a, e), c)) (A^2 -> H(U1(a, e), U1(e, e)),A^2 -> H(U1(a, e), U1(e, e))) (A^2 -> H(U1(a, e), U1(e, k)),A^2 -> H(U1(a, e), U1(e, k))) ---------------------------------------- (1306) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(k, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1307) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(k, c)),A^2 -> H(U1(c, e), U1(k, c))) (A^2 -> H(U1(d, e), U1(k, c)),A^2 -> H(U1(d, e), U1(k, c))) (A^2 -> H(U1(a, e), U1(k, e)),A^2 -> H(U1(a, e), U1(k, e))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) ---------------------------------------- (1308) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(c, e)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1309) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(c, e)),A^2 -> H(U1(c, e), U1(c, e))) (A^2 -> H(U1(d, e), U1(c, e)),A^2 -> H(U1(d, e), U1(c, e))) (A^2 -> H(U1(a, e), U1(e, e)),A^2 -> H(U1(a, e), U1(e, e))) (A^2 -> H(U1(a, e), U1(k, e)),A^2 -> H(U1(a, e), U1(k, e))) ---------------------------------------- (1310) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(c, k)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1311) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(c, k)),A^2 -> H(U1(c, e), U1(c, k))) (A^2 -> H(U1(d, e), U1(c, k)),A^2 -> H(U1(d, e), U1(c, k))) (A^2 -> H(U1(a, e), U1(e, k)),A^2 -> H(U1(a, e), U1(e, k))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) ---------------------------------------- (1312) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(e, e)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1313) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(e, e)),A^2 -> H(U1(c, e), U1(e, e))) (A^2 -> H(U1(d, e), U1(e, e)),A^2 -> H(U1(d, e), U1(e, e))) (A^2 -> H(U1(a, e), e),A^2 -> H(U1(a, e), e)) ---------------------------------------- (1314) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1315) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1316) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(k, d)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1317) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(k, d)),A^2 -> H(U1(c, e), U1(k, d))) (A^2 -> H(U1(d, e), U1(k, d)),A^2 -> H(U1(d, e), U1(k, d))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) ---------------------------------------- (1318) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, e), U1(d, k)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1319) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(d, k)),A^2 -> H(U1(c, e), U1(d, k))) (A^2 -> H(U1(d, e), U1(d, k)),A^2 -> H(U1(d, e), U1(d, k))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) ---------------------------------------- (1320) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(e, b)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1321) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(e, b)),A^2 -> H(U1(c, k), U1(e, b))) (A^2 -> H(U1(d, k), U1(e, b)),A^2 -> H(U1(d, k), U1(e, b))) (A^2 -> H(U1(a, k), b),A^2 -> H(U1(a, k), b)) (A^2 -> H(U1(a, k), U1(e, c)),A^2 -> H(U1(a, k), U1(e, c))) (A^2 -> H(U1(a, k), U1(e, d)),A^2 -> H(U1(a, k), U1(e, d))) ---------------------------------------- (1322) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(k, b)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1323) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(k, b)),A^2 -> H(U1(c, k), U1(k, b))) (A^2 -> H(U1(d, k), U1(k, b)),A^2 -> H(U1(d, k), U1(k, b))) (A^2 -> H(U1(a, k), U1(k, c)),A^2 -> H(U1(a, k), U1(k, c))) (A^2 -> H(U1(a, k), U1(k, d)),A^2 -> H(U1(a, k), U1(k, d))) ---------------------------------------- (1324) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(c, d)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1325) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(c, d)),A^2 -> H(U1(c, k), U1(c, d))) (A^2 -> H(U1(d, k), U1(c, d)),A^2 -> H(U1(d, k), U1(c, d))) (A^2 -> H(U1(a, k), U1(e, d)),A^2 -> H(U1(a, k), U1(e, d))) (A^2 -> H(U1(a, k), U1(k, d)),A^2 -> H(U1(a, k), U1(k, d))) (A^2 -> H(U1(a, k), U1(c, k)),A^2 -> H(U1(a, k), U1(c, k))) ---------------------------------------- (1326) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1327) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(d, c)),A^2 -> H(U1(c, k), U1(d, c))) (A^2 -> H(U1(d, k), U1(d, c)),A^2 -> H(U1(d, k), U1(d, c))) (A^2 -> H(U1(a, k), U1(k, c)),A^2 -> H(U1(a, k), U1(k, c))) (A^2 -> H(U1(a, k), U1(d, e)),A^2 -> H(U1(a, k), U1(d, e))) (A^2 -> H(U1(a, k), U1(d, k)),A^2 -> H(U1(a, k), U1(d, k))) ---------------------------------------- (1328) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(b, e)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1329) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(b, e)),A^2 -> H(U1(c, k), U1(b, e))) (A^2 -> H(U1(d, k), U1(b, e)),A^2 -> H(U1(d, k), U1(b, e))) (A^2 -> H(U1(a, k), U1(c, e)),A^2 -> H(U1(a, k), U1(c, e))) (A^2 -> H(U1(a, k), U1(d, e)),A^2 -> H(U1(a, k), U1(d, e))) ---------------------------------------- (1330) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(b, k)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1331) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(b, k)),A^2 -> H(U1(c, k), U1(b, k))) (A^2 -> H(U1(d, k), U1(b, k)),A^2 -> H(U1(d, k), U1(b, k))) (A^2 -> H(U1(a, k), U1(c, k)),A^2 -> H(U1(a, k), U1(c, k))) (A^2 -> H(U1(a, k), U1(d, k)),A^2 -> H(U1(a, k), U1(d, k))) ---------------------------------------- (1332) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(e, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1333) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(e, c)),A^2 -> H(U1(c, k), U1(e, c))) (A^2 -> H(U1(d, k), U1(e, c)),A^2 -> H(U1(d, k), U1(e, c))) (A^2 -> H(U1(a, k), c),A^2 -> H(U1(a, k), c)) (A^2 -> H(U1(a, k), U1(e, e)),A^2 -> H(U1(a, k), U1(e, e))) (A^2 -> H(U1(a, k), U1(e, k)),A^2 -> H(U1(a, k), U1(e, k))) ---------------------------------------- (1334) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(k, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1335) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(k, c)),A^2 -> H(U1(c, k), U1(k, c))) (A^2 -> H(U1(d, k), U1(k, c)),A^2 -> H(U1(d, k), U1(k, c))) (A^2 -> H(U1(a, k), U1(k, e)),A^2 -> H(U1(a, k), U1(k, e))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (1336) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(c, e)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1337) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(c, e)),A^2 -> H(U1(c, k), U1(c, e))) (A^2 -> H(U1(d, k), U1(c, e)),A^2 -> H(U1(d, k), U1(c, e))) (A^2 -> H(U1(a, k), U1(e, e)),A^2 -> H(U1(a, k), U1(e, e))) (A^2 -> H(U1(a, k), U1(k, e)),A^2 -> H(U1(a, k), U1(k, e))) ---------------------------------------- (1338) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(c, k)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1339) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(c, k)),A^2 -> H(U1(c, k), U1(c, k))) (A^2 -> H(U1(d, k), U1(c, k)),A^2 -> H(U1(d, k), U1(c, k))) (A^2 -> H(U1(a, k), U1(e, k)),A^2 -> H(U1(a, k), U1(e, k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (1340) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(e, e)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1341) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) (A^2 -> H(U1(d, k), U1(e, e)),A^2 -> H(U1(d, k), U1(e, e))) (A^2 -> H(U1(a, k), e),A^2 -> H(U1(a, k), e)) ---------------------------------------- (1342) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1343) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1344) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(k, d)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1345) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(k, d)),A^2 -> H(U1(c, k), U1(k, d))) (A^2 -> H(U1(d, k), U1(k, d)),A^2 -> H(U1(d, k), U1(k, d))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (1346) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, k), U1(d, k)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1347) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(d, k)),A^2 -> H(U1(c, k), U1(d, k))) (A^2 -> H(U1(d, k), U1(d, k)),A^2 -> H(U1(d, k), U1(d, k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (1348) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), b) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1349) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), b),A^2 -> H(U1(c, c), b)) (A^2 -> H(U1(d, c), b),A^2 -> H(U1(d, c), b)) (A^2 -> H(U1(a, e), b),A^2 -> H(U1(a, e), b)) (A^2 -> H(U1(a, k), b),A^2 -> H(U1(a, k), b)) (A^2 -> H(U1(a, c), c),A^2 -> H(U1(a, c), c)) (A^2 -> H(U1(a, c), d),A^2 -> H(U1(a, c), d)) ---------------------------------------- (1350) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), U1(e, d)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1351) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, d)),A^2 -> H(U1(c, c), U1(e, d))) (A^2 -> H(U1(d, c), U1(e, d)),A^2 -> H(U1(d, c), U1(e, d))) (A^2 -> H(U1(a, e), U1(e, d)),A^2 -> H(U1(a, e), U1(e, d))) (A^2 -> H(U1(a, k), U1(e, d)),A^2 -> H(U1(a, k), U1(e, d))) (A^2 -> H(U1(a, c), d),A^2 -> H(U1(a, c), d)) (A^2 -> H(U1(a, c), U1(e, k)),A^2 -> H(U1(a, c), U1(e, k))) ---------------------------------------- (1352) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), U1(d, e)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1353) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(d, e)),A^2 -> H(U1(c, c), U1(d, e))) (A^2 -> H(U1(d, c), U1(d, e)),A^2 -> H(U1(d, c), U1(d, e))) (A^2 -> H(U1(a, e), U1(d, e)),A^2 -> H(U1(a, e), U1(d, e))) (A^2 -> H(U1(a, k), U1(d, e)),A^2 -> H(U1(a, k), U1(d, e))) (A^2 -> H(U1(a, c), U1(k, e)),A^2 -> H(U1(a, c), U1(k, e))) ---------------------------------------- (1354) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), c) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1355) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), c),A^2 -> H(U1(c, c), c)) (A^2 -> H(U1(d, c), c),A^2 -> H(U1(d, c), c)) (A^2 -> H(U1(a, e), c),A^2 -> H(U1(a, e), c)) (A^2 -> H(U1(a, k), c),A^2 -> H(U1(a, k), c)) (A^2 -> H(U1(a, c), e),A^2 -> H(U1(a, c), e)) (A^2 -> H(U1(a, c), k),A^2 -> H(U1(a, c), k)) ---------------------------------------- (1356) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), U1(e, k)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1357) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(e, k)),A^2 -> H(U1(c, c), U1(e, k))) (A^2 -> H(U1(d, c), U1(e, k)),A^2 -> H(U1(d, c), U1(e, k))) (A^2 -> H(U1(a, e), U1(e, k)),A^2 -> H(U1(a, e), U1(e, k))) (A^2 -> H(U1(a, k), U1(e, k)),A^2 -> H(U1(a, k), U1(e, k))) (A^2 -> H(U1(a, c), k),A^2 -> H(U1(a, c), k)) ---------------------------------------- (1358) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), U1(k, e)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1359) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, e)),A^2 -> H(U1(c, c), U1(k, e))) (A^2 -> H(U1(d, c), U1(k, e)),A^2 -> H(U1(d, c), U1(k, e))) (A^2 -> H(U1(a, e), U1(k, e)),A^2 -> H(U1(a, e), U1(k, e))) (A^2 -> H(U1(a, k), U1(k, e)),A^2 -> H(U1(a, k), U1(k, e))) ---------------------------------------- (1360) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), e) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1361) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), e),A^2 -> H(U1(c, c), e)) (A^2 -> H(U1(d, c), e),A^2 -> H(U1(d, c), e)) (A^2 -> H(U1(a, e), e),A^2 -> H(U1(a, e), e)) (A^2 -> H(U1(a, k), e),A^2 -> H(U1(a, k), e)) ---------------------------------------- (1362) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, c), U1(k, k)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1363) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), U1(k, k)),A^2 -> H(U1(c, c), U1(k, k))) (A^2 -> H(U1(d, c), U1(k, k)),A^2 -> H(U1(d, c), U1(k, k))) (A^2 -> H(U1(a, e), U1(k, k)),A^2 -> H(U1(a, e), U1(k, k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (1364) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), b) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1365) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), b),A^2 -> H(U1(c, d), b)) (A^2 -> H(U1(d, d), b),A^2 -> H(U1(d, d), b)) (A^2 -> H(U1(a, k), b),A^2 -> H(U1(a, k), b)) (A^2 -> H(U1(a, d), c),A^2 -> H(U1(a, d), c)) (A^2 -> H(U1(a, d), d),A^2 -> H(U1(a, d), d)) ---------------------------------------- (1366) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), U1(e, d)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1367) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(e, d)),A^2 -> H(U1(c, d), U1(e, d))) (A^2 -> H(U1(d, d), U1(e, d)),A^2 -> H(U1(d, d), U1(e, d))) (A^2 -> H(U1(a, k), U1(e, d)),A^2 -> H(U1(a, k), U1(e, d))) (A^2 -> H(U1(a, d), d),A^2 -> H(U1(a, d), d)) (A^2 -> H(U1(a, d), U1(e, k)),A^2 -> H(U1(a, d), U1(e, k))) ---------------------------------------- (1368) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), U1(d, e)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1369) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(d, e)),A^2 -> H(U1(c, d), U1(d, e))) (A^2 -> H(U1(d, d), U1(d, e)),A^2 -> H(U1(d, d), U1(d, e))) (A^2 -> H(U1(a, k), U1(d, e)),A^2 -> H(U1(a, k), U1(d, e))) (A^2 -> H(U1(a, d), U1(k, e)),A^2 -> H(U1(a, d), U1(k, e))) ---------------------------------------- (1370) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), c) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1371) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), c),A^2 -> H(U1(c, d), c)) (A^2 -> H(U1(d, d), c),A^2 -> H(U1(d, d), c)) (A^2 -> H(U1(a, k), c),A^2 -> H(U1(a, k), c)) (A^2 -> H(U1(a, d), e),A^2 -> H(U1(a, d), e)) (A^2 -> H(U1(a, d), k),A^2 -> H(U1(a, d), k)) ---------------------------------------- (1372) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), U1(e, k)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1373) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(e, k)),A^2 -> H(U1(c, d), U1(e, k))) (A^2 -> H(U1(d, d), U1(e, k)),A^2 -> H(U1(d, d), U1(e, k))) (A^2 -> H(U1(a, k), U1(e, k)),A^2 -> H(U1(a, k), U1(e, k))) (A^2 -> H(U1(a, d), k),A^2 -> H(U1(a, d), k)) ---------------------------------------- (1374) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), U1(k, e)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1375) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(k, e)),A^2 -> H(U1(c, d), U1(k, e))) (A^2 -> H(U1(d, d), U1(k, e)),A^2 -> H(U1(d, d), U1(k, e))) (A^2 -> H(U1(a, k), U1(k, e)),A^2 -> H(U1(a, k), U1(k, e))) ---------------------------------------- (1376) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), e) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1377) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), e),A^2 -> H(U1(c, d), e)) (A^2 -> H(U1(d, d), e),A^2 -> H(U1(d, d), e)) (A^2 -> H(U1(a, k), e),A^2 -> H(U1(a, k), e)) ---------------------------------------- (1378) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, d), U1(k, k)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1379) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), U1(k, k)),A^2 -> H(U1(c, d), U1(k, k))) (A^2 -> H(U1(d, d), U1(k, k)),A^2 -> H(U1(d, d), U1(k, k))) (A^2 -> H(U1(a, k), U1(k, k)),A^2 -> H(U1(a, k), U1(k, k))) ---------------------------------------- (1380) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, a), d) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1381) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), d),A^2 -> H(U1(c, a), d)) (A^2 -> H(U1(d, a), d),A^2 -> H(U1(d, a), d)) (A^2 -> H(U1(a, c), d),A^2 -> H(U1(a, c), d)) (A^2 -> H(U1(a, d), d),A^2 -> H(U1(a, d), d)) (A^2 -> H(U1(a, a), k),A^2 -> H(U1(a, a), k)) ---------------------------------------- (1382) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(a, a), k) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1383) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, a), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, a), k),A^2 -> H(U1(c, a), k)) (A^2 -> H(U1(d, a), k),A^2 -> H(U1(d, a), k)) (A^2 -> H(U1(a, c), k),A^2 -> H(U1(a, c), k)) (A^2 -> H(U1(a, d), k),A^2 -> H(U1(a, d), k)) ---------------------------------------- (1384) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(k, U1(b, b)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1385) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(b, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, b)),A^2 -> H(k, U1(c, b))) (A^2 -> H(k, U1(d, b)),A^2 -> H(k, U1(d, b))) (A^2 -> H(k, U1(b, c)),A^2 -> H(k, U1(b, c))) (A^2 -> H(k, U1(b, d)),A^2 -> H(k, U1(b, d))) ---------------------------------------- (1386) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(k, f(c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1387) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, f(c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, c)),A^2 -> H(k, U1(c, c))) (A^2 -> H(k, f(e)),A^2 -> H(k, f(e))) (A^2 -> H(k, f(k)),A^2 -> H(k, f(k))) ---------------------------------------- (1388) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(k, f(d)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1389) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, f(d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, d)),A^2 -> H(k, U1(d, d))) (A^2 -> H(k, f(k)),A^2 -> H(k, f(k))) ---------------------------------------- (1390) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, U1(c, b)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1391) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, b)),A^2 -> H(e, U1(c, b))) (A^2 -> H(k, U1(c, b)),A^2 -> H(k, U1(c, b))) (A^2 -> H(c, U1(e, b)),A^2 -> H(c, U1(e, b))) (A^2 -> H(c, U1(k, b)),A^2 -> H(c, U1(k, b))) (A^2 -> H(c, U1(c, c)),A^2 -> H(c, U1(c, c))) (A^2 -> H(c, U1(c, d)),A^2 -> H(c, U1(c, d))) ---------------------------------------- (1392) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, U1(d, b)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1393) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, b)),A^2 -> H(e, U1(d, b))) (A^2 -> H(k, U1(d, b)),A^2 -> H(k, U1(d, b))) (A^2 -> H(c, U1(k, b)),A^2 -> H(c, U1(k, b))) (A^2 -> H(c, U1(d, c)),A^2 -> H(c, U1(d, c))) (A^2 -> H(c, U1(d, d)),A^2 -> H(c, U1(d, d))) ---------------------------------------- (1394) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, U1(b, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1395) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, c)),A^2 -> H(e, U1(b, c))) (A^2 -> H(k, U1(b, c)),A^2 -> H(k, U1(b, c))) (A^2 -> H(c, U1(c, c)),A^2 -> H(c, U1(c, c))) (A^2 -> H(c, U1(d, c)),A^2 -> H(c, U1(d, c))) (A^2 -> H(c, U1(b, e)),A^2 -> H(c, U1(b, e))) (A^2 -> H(c, U1(b, k)),A^2 -> H(c, U1(b, k))) ---------------------------------------- (1396) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, U1(b, d)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1397) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, d)),A^2 -> H(e, U1(b, d))) (A^2 -> H(k, U1(b, d)),A^2 -> H(k, U1(b, d))) (A^2 -> H(c, U1(c, d)),A^2 -> H(c, U1(c, d))) (A^2 -> H(c, U1(d, d)),A^2 -> H(c, U1(d, d))) (A^2 -> H(c, U1(b, k)),A^2 -> H(c, U1(b, k))) ---------------------------------------- (1398) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, U1(c, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1399) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, c)),A^2 -> H(e, U1(c, c))) (A^2 -> H(k, U1(c, c)),A^2 -> H(k, U1(c, c))) (A^2 -> H(c, U1(e, c)),A^2 -> H(c, U1(e, c))) (A^2 -> H(c, U1(k, c)),A^2 -> H(c, U1(k, c))) (A^2 -> H(c, U1(c, e)),A^2 -> H(c, U1(c, e))) (A^2 -> H(c, U1(c, k)),A^2 -> H(c, U1(c, k))) ---------------------------------------- (1400) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, f(e)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1401) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(e)),A^2 -> H(e, f(e))) (A^2 -> H(k, f(e)),A^2 -> H(k, f(e))) (A^2 -> H(c, U1(e, e)),A^2 -> H(c, U1(e, e))) ---------------------------------------- (1402) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, f(k)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1403) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, f(k)),A^2 -> H(e, f(k))) (A^2 -> H(k, f(k)),A^2 -> H(k, f(k))) (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) ---------------------------------------- (1404) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(c, U1(d, d)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1405) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, d)),A^2 -> H(e, U1(d, d))) (A^2 -> H(k, U1(d, d)),A^2 -> H(k, U1(d, d))) (A^2 -> H(c, U1(k, d)),A^2 -> H(c, U1(k, d))) (A^2 -> H(c, U1(d, k)),A^2 -> H(c, U1(d, k))) ---------------------------------------- (1406) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), U1(c, b)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1407) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, b)),A^2 -> H(k, U1(c, b))) (A^2 -> H(U1(e, k), U1(e, b)),A^2 -> H(U1(e, k), U1(e, b))) (A^2 -> H(U1(e, k), U1(k, b)),A^2 -> H(U1(e, k), U1(k, b))) (A^2 -> H(U1(e, k), U1(c, c)),A^2 -> H(U1(e, k), U1(c, c))) (A^2 -> H(U1(e, k), U1(c, d)),A^2 -> H(U1(e, k), U1(c, d))) ---------------------------------------- (1408) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), U1(d, b)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1409) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, b)),A^2 -> H(k, U1(d, b))) (A^2 -> H(U1(e, k), U1(k, b)),A^2 -> H(U1(e, k), U1(k, b))) (A^2 -> H(U1(e, k), U1(d, c)),A^2 -> H(U1(e, k), U1(d, c))) (A^2 -> H(U1(e, k), U1(d, d)),A^2 -> H(U1(e, k), U1(d, d))) ---------------------------------------- (1410) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), U1(b, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1411) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, c)),A^2 -> H(k, U1(b, c))) (A^2 -> H(U1(e, k), U1(c, c)),A^2 -> H(U1(e, k), U1(c, c))) (A^2 -> H(U1(e, k), U1(d, c)),A^2 -> H(U1(e, k), U1(d, c))) (A^2 -> H(U1(e, k), U1(b, e)),A^2 -> H(U1(e, k), U1(b, e))) (A^2 -> H(U1(e, k), U1(b, k)),A^2 -> H(U1(e, k), U1(b, k))) ---------------------------------------- (1412) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), U1(b, d)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1413) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, d)),A^2 -> H(k, U1(b, d))) (A^2 -> H(U1(e, k), U1(c, d)),A^2 -> H(U1(e, k), U1(c, d))) (A^2 -> H(U1(e, k), U1(d, d)),A^2 -> H(U1(e, k), U1(d, d))) (A^2 -> H(U1(e, k), U1(b, k)),A^2 -> H(U1(e, k), U1(b, k))) ---------------------------------------- (1414) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), U1(c, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1415) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, c)),A^2 -> H(k, U1(c, c))) (A^2 -> H(U1(e, k), U1(e, c)),A^2 -> H(U1(e, k), U1(e, c))) (A^2 -> H(U1(e, k), U1(k, c)),A^2 -> H(U1(e, k), U1(k, c))) (A^2 -> H(U1(e, k), U1(c, e)),A^2 -> H(U1(e, k), U1(c, e))) (A^2 -> H(U1(e, k), U1(c, k)),A^2 -> H(U1(e, k), U1(c, k))) ---------------------------------------- (1416) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), f(e)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1417) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(e)),A^2 -> H(k, f(e))) (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) ---------------------------------------- (1418) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), f(k)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1419) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(k)),A^2 -> H(k, f(k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (1420) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, k), U1(d, d)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1421) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, d)),A^2 -> H(k, U1(d, d))) (A^2 -> H(U1(e, k), U1(k, d)),A^2 -> H(U1(e, k), U1(k, d))) (A^2 -> H(U1(e, k), U1(d, k)),A^2 -> H(U1(e, k), U1(d, k))) ---------------------------------------- (1422) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, b)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1423) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, b)),A^2 -> H(c, U1(e, b))) (A^2 -> H(U1(e, e), U1(e, b)),A^2 -> H(U1(e, e), U1(e, b))) (A^2 -> H(U1(e, k), U1(e, b)),A^2 -> H(U1(e, k), U1(e, b))) (A^2 -> H(U1(e, c), b),A^2 -> H(U1(e, c), b)) (A^2 -> H(U1(e, c), U1(e, c)),A^2 -> H(U1(e, c), U1(e, c))) (A^2 -> H(U1(e, c), U1(e, d)),A^2 -> H(U1(e, c), U1(e, d))) ---------------------------------------- (1424) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(k, b)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1425) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, b)),A^2 -> H(c, U1(k, b))) (A^2 -> H(U1(e, e), U1(k, b)),A^2 -> H(U1(e, e), U1(k, b))) (A^2 -> H(U1(e, k), U1(k, b)),A^2 -> H(U1(e, k), U1(k, b))) (A^2 -> H(U1(e, c), U1(k, c)),A^2 -> H(U1(e, c), U1(k, c))) (A^2 -> H(U1(e, c), U1(k, d)),A^2 -> H(U1(e, c), U1(k, d))) ---------------------------------------- (1426) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(c, d)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1427) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, d)),A^2 -> H(c, U1(c, d))) (A^2 -> H(U1(e, e), U1(c, d)),A^2 -> H(U1(e, e), U1(c, d))) (A^2 -> H(U1(e, k), U1(c, d)),A^2 -> H(U1(e, k), U1(c, d))) (A^2 -> H(U1(e, c), U1(e, d)),A^2 -> H(U1(e, c), U1(e, d))) (A^2 -> H(U1(e, c), U1(k, d)),A^2 -> H(U1(e, c), U1(k, d))) (A^2 -> H(U1(e, c), U1(c, k)),A^2 -> H(U1(e, c), U1(c, k))) ---------------------------------------- (1428) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1429) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, c)),A^2 -> H(c, U1(d, c))) (A^2 -> H(U1(e, e), U1(d, c)),A^2 -> H(U1(e, e), U1(d, c))) (A^2 -> H(U1(e, k), U1(d, c)),A^2 -> H(U1(e, k), U1(d, c))) (A^2 -> H(U1(e, c), U1(k, c)),A^2 -> H(U1(e, c), U1(k, c))) (A^2 -> H(U1(e, c), U1(d, e)),A^2 -> H(U1(e, c), U1(d, e))) (A^2 -> H(U1(e, c), U1(d, k)),A^2 -> H(U1(e, c), U1(d, k))) ---------------------------------------- (1430) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, e)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1431) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, e)),A^2 -> H(c, U1(b, e))) (A^2 -> H(U1(e, e), U1(b, e)),A^2 -> H(U1(e, e), U1(b, e))) (A^2 -> H(U1(e, k), U1(b, e)),A^2 -> H(U1(e, k), U1(b, e))) (A^2 -> H(U1(e, c), U1(c, e)),A^2 -> H(U1(e, c), U1(c, e))) (A^2 -> H(U1(e, c), U1(d, e)),A^2 -> H(U1(e, c), U1(d, e))) ---------------------------------------- (1432) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(b, k)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1433) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, k)),A^2 -> H(c, U1(b, k))) (A^2 -> H(U1(e, e), U1(b, k)),A^2 -> H(U1(e, e), U1(b, k))) (A^2 -> H(U1(e, k), U1(b, k)),A^2 -> H(U1(e, k), U1(b, k))) (A^2 -> H(U1(e, c), U1(c, k)),A^2 -> H(U1(e, c), U1(c, k))) (A^2 -> H(U1(e, c), U1(d, k)),A^2 -> H(U1(e, c), U1(d, k))) ---------------------------------------- (1434) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1435) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, c)),A^2 -> H(c, U1(k, c))) (A^2 -> H(U1(e, e), U1(k, c)),A^2 -> H(U1(e, e), U1(k, c))) (A^2 -> H(U1(e, k), U1(k, c)),A^2 -> H(U1(e, k), U1(k, c))) (A^2 -> H(U1(e, c), U1(k, e)),A^2 -> H(U1(e, c), U1(k, e))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) ---------------------------------------- (1436) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(c, e)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1437) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, e)),A^2 -> H(c, U1(c, e))) (A^2 -> H(U1(e, e), U1(c, e)),A^2 -> H(U1(e, e), U1(c, e))) (A^2 -> H(U1(e, k), U1(c, e)),A^2 -> H(U1(e, k), U1(c, e))) (A^2 -> H(U1(e, c), U1(e, e)),A^2 -> H(U1(e, c), U1(e, e))) (A^2 -> H(U1(e, c), U1(k, e)),A^2 -> H(U1(e, c), U1(k, e))) ---------------------------------------- (1438) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(c, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1439) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, k)),A^2 -> H(c, U1(c, k))) (A^2 -> H(U1(e, e), U1(c, k)),A^2 -> H(U1(e, e), U1(c, k))) (A^2 -> H(U1(e, k), U1(c, k)),A^2 -> H(U1(e, k), U1(c, k))) (A^2 -> H(U1(e, c), U1(e, k)),A^2 -> H(U1(e, c), U1(e, k))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) ---------------------------------------- (1440) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1441) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, e)),A^2 -> H(c, U1(e, e))) (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(e, c), e),A^2 -> H(U1(e, c), e)) ---------------------------------------- (1442) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1443) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (1444) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(k, d)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1445) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, d)),A^2 -> H(c, U1(k, d))) (A^2 -> H(U1(e, e), U1(k, d)),A^2 -> H(U1(e, e), U1(k, d))) (A^2 -> H(U1(e, k), U1(k, d)),A^2 -> H(U1(e, k), U1(k, d))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) ---------------------------------------- (1446) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(e, c), U1(d, k)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1447) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, k)),A^2 -> H(c, U1(d, k))) (A^2 -> H(U1(e, e), U1(d, k)),A^2 -> H(U1(e, e), U1(d, k))) (A^2 -> H(U1(e, k), U1(d, k)),A^2 -> H(U1(e, k), U1(d, k))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) ---------------------------------------- (1448) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(c, b)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1449) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, b)),A^2 -> H(U1(k, e), U1(e, b))) (A^2 -> H(U1(k, e), U1(k, b)),A^2 -> H(U1(k, e), U1(k, b))) (A^2 -> H(U1(k, e), U1(c, c)),A^2 -> H(U1(k, e), U1(c, c))) (A^2 -> H(U1(k, e), U1(c, d)),A^2 -> H(U1(k, e), U1(c, d))) ---------------------------------------- (1450) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(d, b)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1451) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, b)),A^2 -> H(U1(k, e), U1(k, b))) (A^2 -> H(U1(k, e), U1(d, c)),A^2 -> H(U1(k, e), U1(d, c))) (A^2 -> H(U1(k, e), U1(d, d)),A^2 -> H(U1(k, e), U1(d, d))) ---------------------------------------- (1452) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(b, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1453) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, c)),A^2 -> H(U1(k, e), U1(c, c))) (A^2 -> H(U1(k, e), U1(d, c)),A^2 -> H(U1(k, e), U1(d, c))) (A^2 -> H(U1(k, e), U1(b, e)),A^2 -> H(U1(k, e), U1(b, e))) (A^2 -> H(U1(k, e), U1(b, k)),A^2 -> H(U1(k, e), U1(b, k))) ---------------------------------------- (1454) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(b, d)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1455) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, d)),A^2 -> H(U1(k, e), U1(c, d))) (A^2 -> H(U1(k, e), U1(d, d)),A^2 -> H(U1(k, e), U1(d, d))) (A^2 -> H(U1(k, e), U1(b, k)),A^2 -> H(U1(k, e), U1(b, k))) ---------------------------------------- (1456) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(c, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1457) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, c)),A^2 -> H(U1(k, e), U1(e, c))) (A^2 -> H(U1(k, e), U1(k, c)),A^2 -> H(U1(k, e), U1(k, c))) (A^2 -> H(U1(k, e), U1(c, e)),A^2 -> H(U1(k, e), U1(c, e))) (A^2 -> H(U1(k, e), U1(c, k)),A^2 -> H(U1(k, e), U1(c, k))) ---------------------------------------- (1458) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), f(e)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1459) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) ---------------------------------------- (1460) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), f(k)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1461) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (1462) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1463) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1464) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, e), U1(d, d)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1465) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, d)),A^2 -> H(U1(k, e), U1(k, d))) (A^2 -> H(U1(k, e), U1(d, k)),A^2 -> H(U1(k, e), U1(d, k))) ---------------------------------------- (1466) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(e, b)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1467) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, b)),A^2 -> H(U1(k, e), U1(e, b))) (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(U1(k, c), b),A^2 -> H(U1(k, c), b)) (A^2 -> H(U1(k, c), U1(e, c)),A^2 -> H(U1(k, c), U1(e, c))) (A^2 -> H(U1(k, c), U1(e, d)),A^2 -> H(U1(k, c), U1(e, d))) ---------------------------------------- (1468) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, b)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1469) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, b)),A^2 -> H(U1(k, e), U1(k, b))) (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(k, c), U1(k, c)),A^2 -> H(U1(k, c), U1(k, c))) (A^2 -> H(U1(k, c), U1(k, d)),A^2 -> H(U1(k, c), U1(k, d))) ---------------------------------------- (1470) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(c, d)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1471) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, d)),A^2 -> H(U1(k, e), U1(c, d))) (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(U1(k, c), U1(e, d)),A^2 -> H(U1(k, c), U1(e, d))) (A^2 -> H(U1(k, c), U1(k, d)),A^2 -> H(U1(k, c), U1(k, d))) (A^2 -> H(U1(k, c), U1(c, k)),A^2 -> H(U1(k, c), U1(c, k))) ---------------------------------------- (1472) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(d, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1473) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, c)),A^2 -> H(U1(k, e), U1(d, c))) (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(k, c), U1(k, c)),A^2 -> H(U1(k, c), U1(k, c))) (A^2 -> H(U1(k, c), U1(d, e)),A^2 -> H(U1(k, c), U1(d, e))) (A^2 -> H(U1(k, c), U1(d, k)),A^2 -> H(U1(k, c), U1(d, k))) ---------------------------------------- (1474) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(b, e)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1475) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, e)),A^2 -> H(U1(k, e), U1(b, e))) (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(U1(k, c), U1(c, e)),A^2 -> H(U1(k, c), U1(c, e))) (A^2 -> H(U1(k, c), U1(d, e)),A^2 -> H(U1(k, c), U1(d, e))) ---------------------------------------- (1476) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(b, k)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1477) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, k)),A^2 -> H(U1(k, e), U1(b, k))) (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) (A^2 -> H(U1(k, c), U1(c, k)),A^2 -> H(U1(k, c), U1(c, k))) (A^2 -> H(U1(k, c), U1(d, k)),A^2 -> H(U1(k, c), U1(d, k))) ---------------------------------------- (1478) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1479) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, c)),A^2 -> H(U1(k, e), U1(e, c))) (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(k, c), c),A^2 -> H(U1(k, c), c)) (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) (A^2 -> H(U1(k, c), U1(e, k)),A^2 -> H(U1(k, c), U1(e, k))) ---------------------------------------- (1480) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, e)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1481) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, e)),A^2 -> H(U1(k, e), U1(c, e))) (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) (A^2 -> H(U1(k, c), U1(k, e)),A^2 -> H(U1(k, c), U1(k, e))) ---------------------------------------- (1482) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(c, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1483) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, k)),A^2 -> H(U1(k, e), U1(c, k))) (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) (A^2 -> H(U1(k, c), U1(e, k)),A^2 -> H(U1(k, c), U1(e, k))) (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) ---------------------------------------- (1484) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1485) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, c), e),A^2 -> H(U1(k, c), e)) ---------------------------------------- (1486) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1487) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1488) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1489) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1490) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1491) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1492) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(k, d)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1493) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, d)),A^2 -> H(U1(k, e), U1(k, d))) (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) ---------------------------------------- (1494) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, c), U1(d, k)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1495) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, k)),A^2 -> H(U1(k, e), U1(d, k))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) ---------------------------------------- (1496) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(e, b)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1497) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, b)),A^2 -> H(U1(e, e), U1(e, b))) (A^2 -> H(U1(k, e), U1(e, b)),A^2 -> H(U1(k, e), U1(e, b))) (A^2 -> H(U1(c, e), b),A^2 -> H(U1(c, e), b)) (A^2 -> H(U1(c, e), U1(e, c)),A^2 -> H(U1(c, e), U1(e, c))) (A^2 -> H(U1(c, e), U1(e, d)),A^2 -> H(U1(c, e), U1(e, d))) ---------------------------------------- (1498) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(k, b)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1499) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, b)),A^2 -> H(U1(e, e), U1(k, b))) (A^2 -> H(U1(k, e), U1(k, b)),A^2 -> H(U1(k, e), U1(k, b))) (A^2 -> H(U1(c, e), U1(k, c)),A^2 -> H(U1(c, e), U1(k, c))) (A^2 -> H(U1(c, e), U1(k, d)),A^2 -> H(U1(c, e), U1(k, d))) ---------------------------------------- (1500) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, d)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1501) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, d)),A^2 -> H(U1(e, e), U1(c, d))) (A^2 -> H(U1(k, e), U1(c, d)),A^2 -> H(U1(k, e), U1(c, d))) (A^2 -> H(U1(c, e), U1(e, d)),A^2 -> H(U1(c, e), U1(e, d))) (A^2 -> H(U1(c, e), U1(k, d)),A^2 -> H(U1(c, e), U1(k, d))) (A^2 -> H(U1(c, e), U1(c, k)),A^2 -> H(U1(c, e), U1(c, k))) ---------------------------------------- (1502) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(d, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1503) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, c)),A^2 -> H(U1(e, e), U1(d, c))) (A^2 -> H(U1(k, e), U1(d, c)),A^2 -> H(U1(k, e), U1(d, c))) (A^2 -> H(U1(c, e), U1(k, c)),A^2 -> H(U1(c, e), U1(k, c))) (A^2 -> H(U1(c, e), U1(d, e)),A^2 -> H(U1(c, e), U1(d, e))) (A^2 -> H(U1(c, e), U1(d, k)),A^2 -> H(U1(c, e), U1(d, k))) ---------------------------------------- (1504) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(b, e)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1505) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, e)),A^2 -> H(U1(e, e), U1(b, e))) (A^2 -> H(U1(k, e), U1(b, e)),A^2 -> H(U1(k, e), U1(b, e))) (A^2 -> H(U1(c, e), U1(c, e)),A^2 -> H(U1(c, e), U1(c, e))) (A^2 -> H(U1(c, e), U1(d, e)),A^2 -> H(U1(c, e), U1(d, e))) ---------------------------------------- (1506) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(b, k)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1507) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(b, k)),A^2 -> H(U1(e, e), U1(b, k))) (A^2 -> H(U1(k, e), U1(b, k)),A^2 -> H(U1(k, e), U1(b, k))) (A^2 -> H(U1(c, e), U1(c, k)),A^2 -> H(U1(c, e), U1(c, k))) (A^2 -> H(U1(c, e), U1(d, k)),A^2 -> H(U1(c, e), U1(d, k))) ---------------------------------------- (1508) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(e, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1509) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, c)),A^2 -> H(U1(e, e), U1(e, c))) (A^2 -> H(U1(k, e), U1(e, c)),A^2 -> H(U1(k, e), U1(e, c))) (A^2 -> H(U1(c, e), c),A^2 -> H(U1(c, e), c)) (A^2 -> H(U1(c, e), U1(e, e)),A^2 -> H(U1(c, e), U1(e, e))) (A^2 -> H(U1(c, e), U1(e, k)),A^2 -> H(U1(c, e), U1(e, k))) ---------------------------------------- (1510) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1511) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, c)),A^2 -> H(U1(e, e), U1(k, c))) (A^2 -> H(U1(k, e), U1(k, c)),A^2 -> H(U1(k, e), U1(k, c))) (A^2 -> H(U1(c, e), U1(k, e)),A^2 -> H(U1(c, e), U1(k, e))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) ---------------------------------------- (1512) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(c, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1513) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(c, k)),A^2 -> H(U1(e, e), U1(c, k))) (A^2 -> H(U1(k, e), U1(c, k)),A^2 -> H(U1(k, e), U1(c, k))) (A^2 -> H(U1(c, e), U1(e, k)),A^2 -> H(U1(c, e), U1(e, k))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) ---------------------------------------- (1514) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1515) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(c, e), e),A^2 -> H(U1(c, e), e)) ---------------------------------------- (1516) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1517) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (1518) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1519) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1520) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(k, d)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1521) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, d)),A^2 -> H(U1(e, e), U1(k, d))) (A^2 -> H(U1(k, e), U1(k, d)),A^2 -> H(U1(k, e), U1(k, d))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) ---------------------------------------- (1522) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, e), U1(d, k)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1523) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, k)),A^2 -> H(U1(e, e), U1(d, k))) (A^2 -> H(U1(k, e), U1(d, k)),A^2 -> H(U1(k, e), U1(d, k))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) ---------------------------------------- (1524) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(e, b)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1525) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(e, b)),A^2 -> H(U1(e, k), U1(e, b))) (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(U1(c, k), b),A^2 -> H(U1(c, k), b)) (A^2 -> H(U1(c, k), U1(e, c)),A^2 -> H(U1(c, k), U1(e, c))) (A^2 -> H(U1(c, k), U1(e, d)),A^2 -> H(U1(c, k), U1(e, d))) ---------------------------------------- (1526) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(k, b)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1527) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(k, b)),A^2 -> H(U1(e, k), U1(k, b))) (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(c, k), U1(k, c)),A^2 -> H(U1(c, k), U1(k, c))) (A^2 -> H(U1(c, k), U1(k, d)),A^2 -> H(U1(c, k), U1(k, d))) ---------------------------------------- (1528) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, d)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1529) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(c, d)),A^2 -> H(U1(e, k), U1(c, d))) (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(U1(c, k), U1(e, d)),A^2 -> H(U1(c, k), U1(e, d))) (A^2 -> H(U1(c, k), U1(k, d)),A^2 -> H(U1(c, k), U1(k, d))) (A^2 -> H(U1(c, k), U1(c, k)),A^2 -> H(U1(c, k), U1(c, k))) ---------------------------------------- (1530) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(d, c)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1531) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(d, c)),A^2 -> H(U1(e, k), U1(d, c))) (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(c, k), U1(k, c)),A^2 -> H(U1(c, k), U1(k, c))) (A^2 -> H(U1(c, k), U1(d, e)),A^2 -> H(U1(c, k), U1(d, e))) (A^2 -> H(U1(c, k), U1(d, k)),A^2 -> H(U1(c, k), U1(d, k))) ---------------------------------------- (1532) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(b, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1533) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(b, e)),A^2 -> H(U1(e, k), U1(b, e))) (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(U1(c, k), U1(c, e)),A^2 -> H(U1(c, k), U1(c, e))) (A^2 -> H(U1(c, k), U1(d, e)),A^2 -> H(U1(c, k), U1(d, e))) ---------------------------------------- (1534) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(b, k)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1535) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(b, k)),A^2 -> H(U1(e, k), U1(b, k))) (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) (A^2 -> H(U1(c, k), U1(c, k)),A^2 -> H(U1(c, k), U1(c, k))) (A^2 -> H(U1(c, k), U1(d, k)),A^2 -> H(U1(c, k), U1(d, k))) ---------------------------------------- (1536) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(e, c)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1537) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(e, c)),A^2 -> H(U1(e, k), U1(e, c))) (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(c, k), c),A^2 -> H(U1(c, k), c)) (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) (A^2 -> H(U1(c, k), U1(e, k)),A^2 -> H(U1(c, k), U1(e, k))) ---------------------------------------- (1538) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(k, c)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1539) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(k, c)),A^2 -> H(U1(e, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(c, k), U1(k, e)),A^2 -> H(U1(c, k), U1(k, e))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (1540) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1541) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(c, e)),A^2 -> H(U1(e, k), U1(c, e))) (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) (A^2 -> H(U1(c, k), U1(k, e)),A^2 -> H(U1(c, k), U1(k, e))) ---------------------------------------- (1542) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1543) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) ---------------------------------------- (1544) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1545) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1546) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(k, d)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1547) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(k, d)),A^2 -> H(U1(e, k), U1(k, d))) (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (1548) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, k), U1(d, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1549) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(d, k)),A^2 -> H(U1(e, k), U1(d, k))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (1550) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, c), b) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1551) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), b),A^2 -> H(U1(e, c), b)) (A^2 -> H(U1(k, c), b),A^2 -> H(U1(k, c), b)) (A^2 -> H(U1(c, e), b),A^2 -> H(U1(c, e), b)) (A^2 -> H(U1(c, k), b),A^2 -> H(U1(c, k), b)) (A^2 -> H(U1(c, c), c),A^2 -> H(U1(c, c), c)) (A^2 -> H(U1(c, c), d),A^2 -> H(U1(c, c), d)) ---------------------------------------- (1552) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, c), U1(e, d)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1553) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(e, d)),A^2 -> H(U1(e, c), U1(e, d))) (A^2 -> H(U1(k, c), U1(e, d)),A^2 -> H(U1(k, c), U1(e, d))) (A^2 -> H(U1(c, e), U1(e, d)),A^2 -> H(U1(c, e), U1(e, d))) (A^2 -> H(U1(c, k), U1(e, d)),A^2 -> H(U1(c, k), U1(e, d))) (A^2 -> H(U1(c, c), d),A^2 -> H(U1(c, c), d)) (A^2 -> H(U1(c, c), U1(e, k)),A^2 -> H(U1(c, c), U1(e, k))) ---------------------------------------- (1554) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, c), U1(d, e)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1555) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(d, e)),A^2 -> H(U1(e, c), U1(d, e))) (A^2 -> H(U1(k, c), U1(d, e)),A^2 -> H(U1(k, c), U1(d, e))) (A^2 -> H(U1(c, e), U1(d, e)),A^2 -> H(U1(c, e), U1(d, e))) (A^2 -> H(U1(c, k), U1(d, e)),A^2 -> H(U1(c, k), U1(d, e))) (A^2 -> H(U1(c, c), U1(k, e)),A^2 -> H(U1(c, c), U1(k, e))) ---------------------------------------- (1556) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1557) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), e),A^2 -> H(U1(e, c), e)) (A^2 -> H(U1(k, c), e),A^2 -> H(U1(k, c), e)) (A^2 -> H(U1(c, e), e),A^2 -> H(U1(c, e), e)) (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) ---------------------------------------- (1558) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(U1(k, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1559) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1560) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(c, c), U1(k, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1561) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (1562) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(c, b)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1563) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, b)),A^2 -> H(e, U1(e, b))) (A^2 -> H(e, U1(k, b)),A^2 -> H(e, U1(k, b))) (A^2 -> H(e, U1(c, c)),A^2 -> H(e, U1(c, c))) (A^2 -> H(e, U1(c, d)),A^2 -> H(e, U1(c, d))) ---------------------------------------- (1564) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(k, b)) A^2 -> H(e, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1565) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1566) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(d, b)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1567) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, b)),A^2 -> H(e, U1(k, b))) (A^2 -> H(e, U1(d, c)),A^2 -> H(e, U1(d, c))) (A^2 -> H(e, U1(d, d)),A^2 -> H(e, U1(d, d))) ---------------------------------------- (1568) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(k, b)) A^2 -> H(e, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1569) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1570) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(b, c)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1571) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, c)),A^2 -> H(e, U1(c, c))) (A^2 -> H(e, U1(d, c)),A^2 -> H(e, U1(d, c))) (A^2 -> H(e, U1(b, e)),A^2 -> H(e, U1(b, e))) (A^2 -> H(e, U1(b, k)),A^2 -> H(e, U1(b, k))) ---------------------------------------- (1572) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(b, d)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1573) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, d)),A^2 -> H(e, U1(c, d))) (A^2 -> H(e, U1(d, d)),A^2 -> H(e, U1(d, d))) (A^2 -> H(e, U1(b, k)),A^2 -> H(e, U1(b, k))) ---------------------------------------- (1574) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(c, c)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1575) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, c)),A^2 -> H(e, U1(e, c))) (A^2 -> H(e, U1(k, c)),A^2 -> H(e, U1(k, c))) (A^2 -> H(e, U1(c, e)),A^2 -> H(e, U1(c, e))) (A^2 -> H(e, U1(c, k)),A^2 -> H(e, U1(c, k))) ---------------------------------------- (1576) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(k, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1577) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1578) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, f(e)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1579) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, e)),A^2 -> H(e, U1(e, e))) ---------------------------------------- (1580) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, f(k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1581) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) ---------------------------------------- (1582) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1583) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1584) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(e, U1(d, d)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1585) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, d)),A^2 -> H(e, U1(k, d))) (A^2 -> H(e, U1(d, k)),A^2 -> H(e, U1(d, k))) ---------------------------------------- (1586) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(k, d)) A^2 -> H(e, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1587) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1588) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, b)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1589) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, b)),A^2 -> H(e, U1(e, b))) (A^2 -> H(U1(e, e), b),A^2 -> H(U1(e, e), b)) (A^2 -> H(U1(e, e), U1(e, c)),A^2 -> H(U1(e, e), U1(e, c))) (A^2 -> H(U1(e, e), U1(e, d)),A^2 -> H(U1(e, e), U1(e, d))) ---------------------------------------- (1590) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(k, b)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1591) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, b)),A^2 -> H(e, U1(k, b))) (A^2 -> H(U1(e, e), U1(k, c)),A^2 -> H(U1(e, e), U1(k, c))) (A^2 -> H(U1(e, e), U1(k, d)),A^2 -> H(U1(e, e), U1(k, d))) ---------------------------------------- (1592) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(e, U1(k, b)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1593) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1594) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(c, d)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1595) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, d)),A^2 -> H(e, U1(c, d))) (A^2 -> H(U1(e, e), U1(e, d)),A^2 -> H(U1(e, e), U1(e, d))) (A^2 -> H(U1(e, e), U1(k, d)),A^2 -> H(U1(e, e), U1(k, d))) (A^2 -> H(U1(e, e), U1(c, k)),A^2 -> H(U1(e, e), U1(c, k))) ---------------------------------------- (1596) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(d, c)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1597) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, c)),A^2 -> H(e, U1(d, c))) (A^2 -> H(U1(e, e), U1(k, c)),A^2 -> H(U1(e, e), U1(k, c))) (A^2 -> H(U1(e, e), U1(d, e)),A^2 -> H(U1(e, e), U1(d, e))) (A^2 -> H(U1(e, e), U1(d, k)),A^2 -> H(U1(e, e), U1(d, k))) ---------------------------------------- (1598) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(b, e)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1599) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, e)),A^2 -> H(e, U1(b, e))) (A^2 -> H(U1(e, e), U1(c, e)),A^2 -> H(U1(e, e), U1(c, e))) (A^2 -> H(U1(e, e), U1(d, e)),A^2 -> H(U1(e, e), U1(d, e))) ---------------------------------------- (1600) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(b, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1601) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, k)),A^2 -> H(e, U1(b, k))) (A^2 -> H(U1(e, e), U1(c, k)),A^2 -> H(U1(e, e), U1(c, k))) (A^2 -> H(U1(e, e), U1(d, k)),A^2 -> H(U1(e, e), U1(d, k))) ---------------------------------------- (1602) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, c)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1603) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, c)),A^2 -> H(e, U1(e, c))) (A^2 -> H(U1(e, e), c),A^2 -> H(U1(e, e), c)) (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(U1(e, e), U1(e, k)),A^2 -> H(U1(e, e), U1(e, k))) ---------------------------------------- (1604) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(k, c)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1605) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, c)),A^2 -> H(e, U1(k, c))) (A^2 -> H(U1(e, e), U1(k, e)),A^2 -> H(U1(e, e), U1(k, e))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (1606) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(e, U1(k, c)) A^2 -> H(U1(e, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1607) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1608) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(c, e)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1609) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, e)),A^2 -> H(e, U1(c, e))) (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(U1(e, e), U1(k, e)),A^2 -> H(U1(e, e), U1(k, e))) ---------------------------------------- (1610) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1611) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, k)),A^2 -> H(e, U1(c, k))) (A^2 -> H(U1(e, e), U1(e, k)),A^2 -> H(U1(e, e), U1(e, k))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (1612) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1613) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) ---------------------------------------- (1614) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(e, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1615) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1616) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(k, d)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1617) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, d)),A^2 -> H(e, U1(k, d))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (1618) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(e, U1(k, d)) A^2 -> H(U1(e, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1619) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1620) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(e, e), U1(d, k)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1621) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, k)),A^2 -> H(e, U1(d, k))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (1622) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), b) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1623) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), b),A^2 -> H(U1(e, e), b)) (A^2 -> H(f(e), c),A^2 -> H(f(e), c)) (A^2 -> H(f(e), d),A^2 -> H(f(e), d)) ---------------------------------------- (1624) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), U1(e, d)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1625) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, d)),A^2 -> H(U1(e, e), U1(e, d))) (A^2 -> H(f(e), d),A^2 -> H(f(e), d)) (A^2 -> H(f(e), U1(e, k)),A^2 -> H(f(e), U1(e, k))) ---------------------------------------- (1626) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), U1(d, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1627) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, e)),A^2 -> H(U1(e, e), U1(d, e))) (A^2 -> H(f(e), U1(k, e)),A^2 -> H(f(e), U1(k, e))) ---------------------------------------- (1628) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), c) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1629) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), c),A^2 -> H(U1(e, e), c)) (A^2 -> H(f(e), e),A^2 -> H(f(e), e)) (A^2 -> H(f(e), k),A^2 -> H(f(e), k)) ---------------------------------------- (1630) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), U1(e, k)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1631) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, k)),A^2 -> H(U1(e, e), U1(e, k))) (A^2 -> H(f(e), k),A^2 -> H(f(e), k)) ---------------------------------------- (1632) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), U1(k, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1633) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, e)),A^2 -> H(U1(e, e), U1(k, e))) ---------------------------------------- (1634) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(f(e), U1(k, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1635) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) ---------------------------------------- (1636) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1637) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) ---------------------------------------- (1638) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1639) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) ---------------------------------------- (1640) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1641) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) ---------------------------------------- (1642) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1643) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) ---------------------------------------- (1644) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1645) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) ---------------------------------------- (1646) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1647) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) ---------------------------------------- (1648) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1649) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) ---------------------------------------- (1650) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1651) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1652) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1653) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1654) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1655) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (1656) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1657) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1658) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1659) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1660) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1661) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (1662) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1663) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1664) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1665) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1666) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1667) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1668) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), b) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1669) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(f(k), c),A^2 -> H(f(k), c)) (A^2 -> H(f(k), d),A^2 -> H(f(k), d)) ---------------------------------------- (1670) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), U1(e, d)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1671) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(f(k), d),A^2 -> H(f(k), d)) (A^2 -> H(f(k), U1(e, k)),A^2 -> H(f(k), U1(e, k))) ---------------------------------------- (1672) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), U1(d, e)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1673) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(f(k), U1(k, e)),A^2 -> H(f(k), U1(k, e))) ---------------------------------------- (1674) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), c) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1675) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(f(k), e),A^2 -> H(f(k), e)) (A^2 -> H(f(k), k),A^2 -> H(f(k), k)) ---------------------------------------- (1676) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), U1(e, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1677) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(f(k), k),A^2 -> H(f(k), k)) ---------------------------------------- (1678) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), U1(k, e)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1679) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (1680) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1681) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1682) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(k), e) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1683) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (1684) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1685) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1686) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(f(c), d) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1687) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), d),A^2 -> H(U1(c, c), d)) (A^2 -> H(f(e), d),A^2 -> H(f(e), d)) (A^2 -> H(f(k), d),A^2 -> H(f(k), d)) (A^2 -> H(f(c), k),A^2 -> H(f(c), k)) ---------------------------------------- (1688) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(c, b)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1689) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) ---------------------------------------- (1690) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(d, b)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1691) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) ---------------------------------------- (1692) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, c)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1693) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, c)),A^2 -> H(U1(k, k), U1(c, c))) (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) ---------------------------------------- (1694) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(b, d)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1695) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(U1(k, k), U1(d, d)),A^2 -> H(U1(k, k), U1(d, d))) (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) ---------------------------------------- (1696) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(c, c)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1697) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) ---------------------------------------- (1698) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), f(e)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1699) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) ---------------------------------------- (1700) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), f(k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1701) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1702) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, k), U1(d, d)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1703) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) ---------------------------------------- (1704) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(e, b)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1705) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(U1(k, d), b),A^2 -> H(U1(k, d), b)) (A^2 -> H(U1(k, d), U1(e, c)),A^2 -> H(U1(k, d), U1(e, c))) (A^2 -> H(U1(k, d), U1(e, d)),A^2 -> H(U1(k, d), U1(e, d))) ---------------------------------------- (1706) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, b)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1707) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(k, d), U1(k, c)),A^2 -> H(U1(k, d), U1(k, c))) (A^2 -> H(U1(k, d), U1(k, d)),A^2 -> H(U1(k, d), U1(k, d))) ---------------------------------------- (1708) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(c, d)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1709) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(U1(k, d), U1(e, d)),A^2 -> H(U1(k, d), U1(e, d))) (A^2 -> H(U1(k, d), U1(k, d)),A^2 -> H(U1(k, d), U1(k, d))) (A^2 -> H(U1(k, d), U1(c, k)),A^2 -> H(U1(k, d), U1(c, k))) ---------------------------------------- (1710) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(d, c)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1711) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(k, d), U1(k, c)),A^2 -> H(U1(k, d), U1(k, c))) (A^2 -> H(U1(k, d), U1(d, e)),A^2 -> H(U1(k, d), U1(d, e))) (A^2 -> H(U1(k, d), U1(d, k)),A^2 -> H(U1(k, d), U1(d, k))) ---------------------------------------- (1712) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(b, e)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1713) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(U1(k, d), U1(c, e)),A^2 -> H(U1(k, d), U1(c, e))) (A^2 -> H(U1(k, d), U1(d, e)),A^2 -> H(U1(k, d), U1(d, e))) ---------------------------------------- (1714) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(b, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1715) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) (A^2 -> H(U1(k, d), U1(c, k)),A^2 -> H(U1(k, d), U1(c, k))) (A^2 -> H(U1(k, d), U1(d, k)),A^2 -> H(U1(k, d), U1(d, k))) ---------------------------------------- (1716) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(e, c)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1717) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(k, d), c),A^2 -> H(U1(k, d), c)) (A^2 -> H(U1(k, d), U1(e, e)),A^2 -> H(U1(k, d), U1(e, e))) (A^2 -> H(U1(k, d), U1(e, k)),A^2 -> H(U1(k, d), U1(e, k))) ---------------------------------------- (1718) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, c)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1719) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, d), U1(k, e)),A^2 -> H(U1(k, d), U1(k, e))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) ---------------------------------------- (1720) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(c, e)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1721) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(k, d), U1(e, e)),A^2 -> H(U1(k, d), U1(e, e))) (A^2 -> H(U1(k, d), U1(k, e)),A^2 -> H(U1(k, d), U1(k, e))) ---------------------------------------- (1722) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(c, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1723) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) (A^2 -> H(U1(k, d), U1(e, k)),A^2 -> H(U1(k, d), U1(e, k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) ---------------------------------------- (1724) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(e, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1725) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, d), e),A^2 -> H(U1(k, d), e)) ---------------------------------------- (1726) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1727) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1728) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1729) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1730) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(k, d), U1(d, k)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1731) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) ---------------------------------------- (1732) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(e, b)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1733) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, b)),A^2 -> H(U1(k, k), U1(e, b))) (A^2 -> H(U1(d, k), b),A^2 -> H(U1(d, k), b)) (A^2 -> H(U1(d, k), U1(e, c)),A^2 -> H(U1(d, k), U1(e, c))) (A^2 -> H(U1(d, k), U1(e, d)),A^2 -> H(U1(d, k), U1(e, d))) ---------------------------------------- (1734) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(k, b)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1735) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, b)),A^2 -> H(U1(k, k), U1(k, b))) (A^2 -> H(U1(d, k), U1(k, c)),A^2 -> H(U1(d, k), U1(k, c))) (A^2 -> H(U1(d, k), U1(k, d)),A^2 -> H(U1(d, k), U1(k, d))) ---------------------------------------- (1736) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(c, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1737) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, d)),A^2 -> H(U1(k, k), U1(c, d))) (A^2 -> H(U1(d, k), U1(e, d)),A^2 -> H(U1(d, k), U1(e, d))) (A^2 -> H(U1(d, k), U1(k, d)),A^2 -> H(U1(d, k), U1(k, d))) (A^2 -> H(U1(d, k), U1(c, k)),A^2 -> H(U1(d, k), U1(c, k))) ---------------------------------------- (1738) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, c)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1739) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, c)),A^2 -> H(U1(k, k), U1(d, c))) (A^2 -> H(U1(d, k), U1(k, c)),A^2 -> H(U1(d, k), U1(k, c))) (A^2 -> H(U1(d, k), U1(d, e)),A^2 -> H(U1(d, k), U1(d, e))) (A^2 -> H(U1(d, k), U1(d, k)),A^2 -> H(U1(d, k), U1(d, k))) ---------------------------------------- (1740) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(b, e)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1741) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, e)),A^2 -> H(U1(k, k), U1(b, e))) (A^2 -> H(U1(d, k), U1(c, e)),A^2 -> H(U1(d, k), U1(c, e))) (A^2 -> H(U1(d, k), U1(d, e)),A^2 -> H(U1(d, k), U1(d, e))) ---------------------------------------- (1742) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(b, k)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1743) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(b, k)),A^2 -> H(U1(k, k), U1(b, k))) (A^2 -> H(U1(d, k), U1(c, k)),A^2 -> H(U1(d, k), U1(c, k))) (A^2 -> H(U1(d, k), U1(d, k)),A^2 -> H(U1(d, k), U1(d, k))) ---------------------------------------- (1744) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(e, c)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1745) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(d, k), c),A^2 -> H(U1(d, k), c)) (A^2 -> H(U1(d, k), U1(e, e)),A^2 -> H(U1(d, k), U1(e, e))) (A^2 -> H(U1(d, k), U1(e, k)),A^2 -> H(U1(d, k), U1(e, k))) ---------------------------------------- (1746) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(k, c)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1747) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(d, k), U1(k, e)),A^2 -> H(U1(d, k), U1(k, e))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1748) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(c, e)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1749) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(d, k), U1(e, e)),A^2 -> H(U1(d, k), U1(e, e))) (A^2 -> H(U1(d, k), U1(k, e)),A^2 -> H(U1(d, k), U1(k, e))) ---------------------------------------- (1750) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(c, k)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1751) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) (A^2 -> H(U1(d, k), U1(e, k)),A^2 -> H(U1(d, k), U1(e, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1752) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(e, e)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1753) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(d, k), e),A^2 -> H(U1(d, k), e)) ---------------------------------------- (1754) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1755) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1756) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1757) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1758) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), b) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1759) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), b),A^2 -> H(U1(k, d), b)) (A^2 -> H(U1(d, k), b),A^2 -> H(U1(d, k), b)) (A^2 -> H(U1(d, d), c),A^2 -> H(U1(d, d), c)) (A^2 -> H(U1(d, d), d),A^2 -> H(U1(d, d), d)) ---------------------------------------- (1760) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), U1(e, d)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1761) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(e, d)),A^2 -> H(U1(k, d), U1(e, d))) (A^2 -> H(U1(d, k), U1(e, d)),A^2 -> H(U1(d, k), U1(e, d))) (A^2 -> H(U1(d, d), d),A^2 -> H(U1(d, d), d)) (A^2 -> H(U1(d, d), U1(e, k)),A^2 -> H(U1(d, d), U1(e, k))) ---------------------------------------- (1762) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), U1(d, e)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1763) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(d, e)),A^2 -> H(U1(k, d), U1(d, e))) (A^2 -> H(U1(d, k), U1(d, e)),A^2 -> H(U1(d, k), U1(d, e))) (A^2 -> H(U1(d, d), U1(k, e)),A^2 -> H(U1(d, d), U1(k, e))) ---------------------------------------- (1764) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), c) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1765) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), c),A^2 -> H(U1(k, d), c)) (A^2 -> H(U1(d, k), c),A^2 -> H(U1(d, k), c)) (A^2 -> H(U1(d, d), e),A^2 -> H(U1(d, d), e)) (A^2 -> H(U1(d, d), k),A^2 -> H(U1(d, d), k)) ---------------------------------------- (1766) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), U1(e, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1767) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(e, k)),A^2 -> H(U1(k, d), U1(e, k))) (A^2 -> H(U1(d, k), U1(e, k)),A^2 -> H(U1(d, k), U1(e, k))) (A^2 -> H(U1(d, d), k),A^2 -> H(U1(d, d), k)) ---------------------------------------- (1768) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), U1(k, e)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1769) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(k, e)),A^2 -> H(U1(k, d), U1(k, e))) (A^2 -> H(U1(d, k), U1(k, e)),A^2 -> H(U1(d, k), U1(k, e))) ---------------------------------------- (1770) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), e) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1771) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), e),A^2 -> H(U1(k, d), e)) (A^2 -> H(U1(d, k), e),A^2 -> H(U1(d, k), e)) ---------------------------------------- (1772) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(k, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1773) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1774) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(d), d) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1775) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), d),A^2 -> H(U1(d, d), d)) (A^2 -> H(f(k), d),A^2 -> H(f(k), d)) (A^2 -> H(f(d), k),A^2 -> H(f(d), k)) ---------------------------------------- (1776) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(d), k) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1777) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(d), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(d, d), k),A^2 -> H(U1(d, d), k)) (A^2 -> H(f(k), k),A^2 -> H(f(k), k)) ---------------------------------------- (1778) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), c) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1779) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), c),A^2 -> H(U1(e, c), c)) (A^2 -> H(U1(k, c), c),A^2 -> H(U1(k, c), c)) (A^2 -> H(U1(c, e), c),A^2 -> H(U1(c, e), c)) (A^2 -> H(U1(c, k), c),A^2 -> H(U1(c, k), c)) (A^2 -> H(U1(c, c), e),A^2 -> H(U1(c, c), e)) (A^2 -> H(U1(c, c), k),A^2 -> H(U1(c, c), k)) ---------------------------------------- (1780) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), U1(e, e)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1781) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(e, e)),A^2 -> H(U1(e, c), U1(e, e))) (A^2 -> H(U1(k, c), U1(e, e)),A^2 -> H(U1(k, c), U1(e, e))) (A^2 -> H(U1(c, e), U1(e, e)),A^2 -> H(U1(c, e), U1(e, e))) (A^2 -> H(U1(c, k), U1(e, e)),A^2 -> H(U1(c, k), U1(e, e))) (A^2 -> H(U1(c, c), e),A^2 -> H(U1(c, c), e)) ---------------------------------------- (1782) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), U1(e, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1783) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(e, k)),A^2 -> H(U1(e, c), U1(e, k))) (A^2 -> H(U1(k, c), U1(e, k)),A^2 -> H(U1(k, c), U1(e, k))) (A^2 -> H(U1(c, e), U1(e, k)),A^2 -> H(U1(c, e), U1(e, k))) (A^2 -> H(U1(c, k), U1(e, k)),A^2 -> H(U1(c, k), U1(e, k))) (A^2 -> H(U1(c, c), k),A^2 -> H(U1(c, c), k)) ---------------------------------------- (1784) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(c), k) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1785) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(c), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), k),A^2 -> H(U1(c, c), k)) (A^2 -> H(f(e), k),A^2 -> H(f(e), k)) (A^2 -> H(f(k), k),A^2 -> H(f(k), k)) ---------------------------------------- (1786) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), U1(k, e)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1787) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), U1(k, e)),A^2 -> H(U1(e, c), U1(k, e))) (A^2 -> H(U1(k, c), U1(k, e)),A^2 -> H(U1(k, c), U1(k, e))) (A^2 -> H(U1(c, e), U1(k, e)),A^2 -> H(U1(c, e), U1(k, e))) (A^2 -> H(U1(c, k), U1(k, e)),A^2 -> H(U1(c, k), U1(k, e))) ---------------------------------------- (1788) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(k), U1(k, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1789) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (1790) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(e), e) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1791) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) ---------------------------------------- (1792) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), U1(k, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1793) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1794) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, b)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1795) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, b)),A^2 -> H(k, U1(c, b))) (A^2 -> H(d, U1(e, b)),A^2 -> H(d, U1(e, b))) (A^2 -> H(d, U1(k, b)),A^2 -> H(d, U1(k, b))) (A^2 -> H(d, U1(c, c)),A^2 -> H(d, U1(c, c))) (A^2 -> H(d, U1(c, d)),A^2 -> H(d, U1(c, d))) ---------------------------------------- (1796) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(d, b)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1797) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, b)),A^2 -> H(k, U1(d, b))) (A^2 -> H(d, U1(k, b)),A^2 -> H(d, U1(k, b))) (A^2 -> H(d, U1(d, c)),A^2 -> H(d, U1(d, c))) (A^2 -> H(d, U1(d, d)),A^2 -> H(d, U1(d, d))) ---------------------------------------- (1798) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(b, c)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1799) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, c)),A^2 -> H(k, U1(b, c))) (A^2 -> H(d, U1(c, c)),A^2 -> H(d, U1(c, c))) (A^2 -> H(d, U1(d, c)),A^2 -> H(d, U1(d, c))) (A^2 -> H(d, U1(b, e)),A^2 -> H(d, U1(b, e))) (A^2 -> H(d, U1(b, k)),A^2 -> H(d, U1(b, k))) ---------------------------------------- (1800) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(b, d)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1801) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, d)),A^2 -> H(k, U1(b, d))) (A^2 -> H(d, U1(c, d)),A^2 -> H(d, U1(c, d))) (A^2 -> H(d, U1(d, d)),A^2 -> H(d, U1(d, d))) (A^2 -> H(d, U1(b, k)),A^2 -> H(d, U1(b, k))) ---------------------------------------- (1802) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, c)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1803) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, c)),A^2 -> H(k, U1(c, c))) (A^2 -> H(d, U1(e, c)),A^2 -> H(d, U1(e, c))) (A^2 -> H(d, U1(k, c)),A^2 -> H(d, U1(k, c))) (A^2 -> H(d, U1(c, e)),A^2 -> H(d, U1(c, e))) (A^2 -> H(d, U1(c, k)),A^2 -> H(d, U1(c, k))) ---------------------------------------- (1804) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, f(e)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1805) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(e)),A^2 -> H(k, f(e))) (A^2 -> H(d, U1(e, e)),A^2 -> H(d, U1(e, e))) ---------------------------------------- (1806) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, f(k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1807) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, f(k)),A^2 -> H(k, f(k))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (1808) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(d, d)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1809) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, d)),A^2 -> H(k, U1(d, d))) (A^2 -> H(d, U1(k, d)),A^2 -> H(d, U1(k, d))) (A^2 -> H(d, U1(d, k)),A^2 -> H(d, U1(d, k))) ---------------------------------------- (1810) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(e, b)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1811) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, b)),A^2 -> H(c, U1(e, b))) (A^2 -> H(d, U1(e, b)),A^2 -> H(d, U1(e, b))) (A^2 -> H(a, b),A^2 -> H(a, b)) (A^2 -> H(a, U1(e, c)),A^2 -> H(a, U1(e, c))) (A^2 -> H(a, U1(e, d)),A^2 -> H(a, U1(e, d))) ---------------------------------------- (1812) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(k, b)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1813) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, b)),A^2 -> H(c, U1(k, b))) (A^2 -> H(d, U1(k, b)),A^2 -> H(d, U1(k, b))) (A^2 -> H(a, U1(k, c)),A^2 -> H(a, U1(k, c))) (A^2 -> H(a, U1(k, d)),A^2 -> H(a, U1(k, d))) ---------------------------------------- (1814) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(c, d)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1815) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, d)),A^2 -> H(c, U1(c, d))) (A^2 -> H(d, U1(c, d)),A^2 -> H(d, U1(c, d))) (A^2 -> H(a, U1(e, d)),A^2 -> H(a, U1(e, d))) (A^2 -> H(a, U1(k, d)),A^2 -> H(a, U1(k, d))) (A^2 -> H(a, U1(c, k)),A^2 -> H(a, U1(c, k))) ---------------------------------------- (1816) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(d, c)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1817) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, c)),A^2 -> H(c, U1(d, c))) (A^2 -> H(d, U1(d, c)),A^2 -> H(d, U1(d, c))) (A^2 -> H(a, U1(k, c)),A^2 -> H(a, U1(k, c))) (A^2 -> H(a, U1(d, e)),A^2 -> H(a, U1(d, e))) (A^2 -> H(a, U1(d, k)),A^2 -> H(a, U1(d, k))) ---------------------------------------- (1818) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(b, e)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1819) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, e)),A^2 -> H(c, U1(b, e))) (A^2 -> H(d, U1(b, e)),A^2 -> H(d, U1(b, e))) (A^2 -> H(a, U1(c, e)),A^2 -> H(a, U1(c, e))) (A^2 -> H(a, U1(d, e)),A^2 -> H(a, U1(d, e))) ---------------------------------------- (1820) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(b, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1821) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(b, k)),A^2 -> H(c, U1(b, k))) (A^2 -> H(d, U1(b, k)),A^2 -> H(d, U1(b, k))) (A^2 -> H(a, U1(c, k)),A^2 -> H(a, U1(c, k))) (A^2 -> H(a, U1(d, k)),A^2 -> H(a, U1(d, k))) ---------------------------------------- (1822) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(e, c)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1823) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, c)),A^2 -> H(c, U1(e, c))) (A^2 -> H(d, U1(e, c)),A^2 -> H(d, U1(e, c))) (A^2 -> H(a, c),A^2 -> H(a, c)) (A^2 -> H(a, U1(e, e)),A^2 -> H(a, U1(e, e))) (A^2 -> H(a, U1(e, k)),A^2 -> H(a, U1(e, k))) ---------------------------------------- (1824) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(k, c)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1825) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, c)),A^2 -> H(c, U1(k, c))) (A^2 -> H(d, U1(k, c)),A^2 -> H(d, U1(k, c))) (A^2 -> H(a, U1(k, e)),A^2 -> H(a, U1(k, e))) (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) ---------------------------------------- (1826) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(c, e)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1827) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, e)),A^2 -> H(c, U1(c, e))) (A^2 -> H(d, U1(c, e)),A^2 -> H(d, U1(c, e))) (A^2 -> H(a, U1(e, e)),A^2 -> H(a, U1(e, e))) (A^2 -> H(a, U1(k, e)),A^2 -> H(a, U1(k, e))) ---------------------------------------- (1828) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(c, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1829) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(c, k)),A^2 -> H(c, U1(c, k))) (A^2 -> H(d, U1(c, k)),A^2 -> H(d, U1(c, k))) (A^2 -> H(a, U1(e, k)),A^2 -> H(a, U1(e, k))) (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) ---------------------------------------- (1830) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(e, e)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1831) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, e)),A^2 -> H(c, U1(e, e))) (A^2 -> H(d, U1(e, e)),A^2 -> H(d, U1(e, e))) (A^2 -> H(a, e),A^2 -> H(a, e)) ---------------------------------------- (1832) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(k, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1833) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (1834) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(k, d)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1835) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, d)),A^2 -> H(c, U1(k, d))) (A^2 -> H(d, U1(k, d)),A^2 -> H(d, U1(k, d))) (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) ---------------------------------------- (1836) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1837) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, k)),A^2 -> H(c, U1(d, k))) (A^2 -> H(d, U1(d, k)),A^2 -> H(d, U1(d, k))) (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) ---------------------------------------- (1838) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, b)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1839) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(e, b)),A^2 -> H(d, U1(e, b))) (A^2 -> H(U1(e, k), U1(e, b)),A^2 -> H(U1(e, k), U1(e, b))) (A^2 -> H(U1(e, d), b),A^2 -> H(U1(e, d), b)) (A^2 -> H(U1(e, d), U1(e, c)),A^2 -> H(U1(e, d), U1(e, c))) (A^2 -> H(U1(e, d), U1(e, d)),A^2 -> H(U1(e, d), U1(e, d))) ---------------------------------------- (1840) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(k, b)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1841) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(k, b)),A^2 -> H(d, U1(k, b))) (A^2 -> H(U1(e, k), U1(k, b)),A^2 -> H(U1(e, k), U1(k, b))) (A^2 -> H(U1(e, d), U1(k, c)),A^2 -> H(U1(e, d), U1(k, c))) (A^2 -> H(U1(e, d), U1(k, d)),A^2 -> H(U1(e, d), U1(k, d))) ---------------------------------------- (1842) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(c, d)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1843) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(c, d)),A^2 -> H(d, U1(c, d))) (A^2 -> H(U1(e, k), U1(c, d)),A^2 -> H(U1(e, k), U1(c, d))) (A^2 -> H(U1(e, d), U1(e, d)),A^2 -> H(U1(e, d), U1(e, d))) (A^2 -> H(U1(e, d), U1(k, d)),A^2 -> H(U1(e, d), U1(k, d))) (A^2 -> H(U1(e, d), U1(c, k)),A^2 -> H(U1(e, d), U1(c, k))) ---------------------------------------- (1844) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(d, c)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1845) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(d, c)),A^2 -> H(d, U1(d, c))) (A^2 -> H(U1(e, k), U1(d, c)),A^2 -> H(U1(e, k), U1(d, c))) (A^2 -> H(U1(e, d), U1(k, c)),A^2 -> H(U1(e, d), U1(k, c))) (A^2 -> H(U1(e, d), U1(d, e)),A^2 -> H(U1(e, d), U1(d, e))) (A^2 -> H(U1(e, d), U1(d, k)),A^2 -> H(U1(e, d), U1(d, k))) ---------------------------------------- (1846) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(b, e)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1847) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(b, e)),A^2 -> H(d, U1(b, e))) (A^2 -> H(U1(e, k), U1(b, e)),A^2 -> H(U1(e, k), U1(b, e))) (A^2 -> H(U1(e, d), U1(c, e)),A^2 -> H(U1(e, d), U1(c, e))) (A^2 -> H(U1(e, d), U1(d, e)),A^2 -> H(U1(e, d), U1(d, e))) ---------------------------------------- (1848) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(b, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1849) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(b, k)),A^2 -> H(d, U1(b, k))) (A^2 -> H(U1(e, k), U1(b, k)),A^2 -> H(U1(e, k), U1(b, k))) (A^2 -> H(U1(e, d), U1(c, k)),A^2 -> H(U1(e, d), U1(c, k))) (A^2 -> H(U1(e, d), U1(d, k)),A^2 -> H(U1(e, d), U1(d, k))) ---------------------------------------- (1850) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, c)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1851) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(e, c)),A^2 -> H(d, U1(e, c))) (A^2 -> H(U1(e, k), U1(e, c)),A^2 -> H(U1(e, k), U1(e, c))) (A^2 -> H(U1(e, d), c),A^2 -> H(U1(e, d), c)) (A^2 -> H(U1(e, d), U1(e, e)),A^2 -> H(U1(e, d), U1(e, e))) (A^2 -> H(U1(e, d), U1(e, k)),A^2 -> H(U1(e, d), U1(e, k))) ---------------------------------------- (1852) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(k, c)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1853) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(k, c)),A^2 -> H(d, U1(k, c))) (A^2 -> H(U1(e, k), U1(k, c)),A^2 -> H(U1(e, k), U1(k, c))) (A^2 -> H(U1(e, d), U1(k, e)),A^2 -> H(U1(e, d), U1(k, e))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1854) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(c, e)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1855) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(c, e)),A^2 -> H(d, U1(c, e))) (A^2 -> H(U1(e, k), U1(c, e)),A^2 -> H(U1(e, k), U1(c, e))) (A^2 -> H(U1(e, d), U1(e, e)),A^2 -> H(U1(e, d), U1(e, e))) (A^2 -> H(U1(e, d), U1(k, e)),A^2 -> H(U1(e, d), U1(k, e))) ---------------------------------------- (1856) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(c, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1857) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(c, k)),A^2 -> H(d, U1(c, k))) (A^2 -> H(U1(e, k), U1(c, k)),A^2 -> H(U1(e, k), U1(c, k))) (A^2 -> H(U1(e, d), U1(e, k)),A^2 -> H(U1(e, d), U1(e, k))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1858) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, e)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1859) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(e, e)),A^2 -> H(d, U1(e, e))) (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(e, d), e),A^2 -> H(U1(e, d), e)) ---------------------------------------- (1860) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1861) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (1862) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(k, d)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1863) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(k, d)),A^2 -> H(d, U1(k, d))) (A^2 -> H(U1(e, k), U1(k, d)),A^2 -> H(U1(e, k), U1(k, d))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1864) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1865) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(d, k)),A^2 -> H(d, U1(d, k))) (A^2 -> H(U1(e, k), U1(d, k)),A^2 -> H(U1(e, k), U1(d, k))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1866) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), b) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1867) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, b),A^2 -> H(a, b)) (A^2 -> H(U1(e, c), b),A^2 -> H(U1(e, c), b)) (A^2 -> H(U1(e, d), b),A^2 -> H(U1(e, d), b)) (A^2 -> H(U1(e, a), c),A^2 -> H(U1(e, a), c)) (A^2 -> H(U1(e, a), d),A^2 -> H(U1(e, a), d)) ---------------------------------------- (1868) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), U1(e, d)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1869) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(e, d)),A^2 -> H(a, U1(e, d))) (A^2 -> H(U1(e, c), U1(e, d)),A^2 -> H(U1(e, c), U1(e, d))) (A^2 -> H(U1(e, d), U1(e, d)),A^2 -> H(U1(e, d), U1(e, d))) (A^2 -> H(U1(e, a), d),A^2 -> H(U1(e, a), d)) (A^2 -> H(U1(e, a), U1(e, k)),A^2 -> H(U1(e, a), U1(e, k))) ---------------------------------------- (1870) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), U1(d, e)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1871) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(d, e)),A^2 -> H(a, U1(d, e))) (A^2 -> H(U1(e, c), U1(d, e)),A^2 -> H(U1(e, c), U1(d, e))) (A^2 -> H(U1(e, d), U1(d, e)),A^2 -> H(U1(e, d), U1(d, e))) (A^2 -> H(U1(e, a), U1(k, e)),A^2 -> H(U1(e, a), U1(k, e))) ---------------------------------------- (1872) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), c) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1873) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, c),A^2 -> H(a, c)) (A^2 -> H(U1(e, c), c),A^2 -> H(U1(e, c), c)) (A^2 -> H(U1(e, d), c),A^2 -> H(U1(e, d), c)) (A^2 -> H(U1(e, a), e),A^2 -> H(U1(e, a), e)) (A^2 -> H(U1(e, a), k),A^2 -> H(U1(e, a), k)) ---------------------------------------- (1874) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), U1(e, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1875) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(e, k)),A^2 -> H(a, U1(e, k))) (A^2 -> H(U1(e, c), U1(e, k)),A^2 -> H(U1(e, c), U1(e, k))) (A^2 -> H(U1(e, d), U1(e, k)),A^2 -> H(U1(e, d), U1(e, k))) (A^2 -> H(U1(e, a), k),A^2 -> H(U1(e, a), k)) ---------------------------------------- (1876) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), U1(k, e)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1877) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(k, e)),A^2 -> H(a, U1(k, e))) (A^2 -> H(U1(e, c), U1(k, e)),A^2 -> H(U1(e, c), U1(k, e))) (A^2 -> H(U1(e, d), U1(k, e)),A^2 -> H(U1(e, d), U1(k, e))) ---------------------------------------- (1878) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), e) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1879) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, e),A^2 -> H(a, e)) (A^2 -> H(U1(e, c), e),A^2 -> H(U1(e, c), e)) (A^2 -> H(U1(e, d), e),A^2 -> H(U1(e, d), e)) ---------------------------------------- (1880) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, a), U1(k, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1881) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, U1(k, k)),A^2 -> H(a, U1(k, k))) (A^2 -> H(U1(e, c), U1(k, k)),A^2 -> H(U1(e, c), U1(k, k))) (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) ---------------------------------------- (1882) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), b) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1883) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), b),A^2 -> H(U1(k, c), b)) (A^2 -> H(U1(k, d), b),A^2 -> H(U1(k, d), b)) (A^2 -> H(U1(k, a), c),A^2 -> H(U1(k, a), c)) (A^2 -> H(U1(k, a), d),A^2 -> H(U1(k, a), d)) ---------------------------------------- (1884) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(e, d)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1885) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, d)),A^2 -> H(U1(k, c), U1(e, d))) (A^2 -> H(U1(k, d), U1(e, d)),A^2 -> H(U1(k, d), U1(e, d))) (A^2 -> H(U1(k, a), d),A^2 -> H(U1(k, a), d)) (A^2 -> H(U1(k, a), U1(e, k)),A^2 -> H(U1(k, a), U1(e, k))) ---------------------------------------- (1886) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(d, e)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1887) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, e)),A^2 -> H(U1(k, c), U1(d, e))) (A^2 -> H(U1(k, d), U1(d, e)),A^2 -> H(U1(k, d), U1(d, e))) (A^2 -> H(U1(k, a), U1(k, e)),A^2 -> H(U1(k, a), U1(k, e))) ---------------------------------------- (1888) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), c) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1889) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), c),A^2 -> H(U1(k, c), c)) (A^2 -> H(U1(k, d), c),A^2 -> H(U1(k, d), c)) (A^2 -> H(U1(k, a), e),A^2 -> H(U1(k, a), e)) (A^2 -> H(U1(k, a), k),A^2 -> H(U1(k, a), k)) ---------------------------------------- (1890) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(k, a), e) A^2 -> H(U1(k, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1891) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (1892) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(e, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1893) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, k)),A^2 -> H(U1(k, c), U1(e, k))) (A^2 -> H(U1(k, d), U1(e, k)),A^2 -> H(U1(k, d), U1(e, k))) (A^2 -> H(U1(k, a), k),A^2 -> H(U1(k, a), k)) ---------------------------------------- (1894) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(k, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1895) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1896) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(k, e)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1897) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, e)),A^2 -> H(U1(k, c), U1(k, e))) (A^2 -> H(U1(k, d), U1(k, e)),A^2 -> H(U1(k, d), U1(k, e))) ---------------------------------------- (1898) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, a), U1(k, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1899) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) ---------------------------------------- (1900) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), b) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1901) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), b),A^2 -> H(U1(e, d), b)) (A^2 -> H(U1(k, d), b),A^2 -> H(U1(k, d), b)) (A^2 -> H(U1(c, k), b),A^2 -> H(U1(c, k), b)) (A^2 -> H(U1(c, d), c),A^2 -> H(U1(c, d), c)) (A^2 -> H(U1(c, d), d),A^2 -> H(U1(c, d), d)) ---------------------------------------- (1902) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), U1(e, d)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1903) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(e, d)),A^2 -> H(U1(e, d), U1(e, d))) (A^2 -> H(U1(k, d), U1(e, d)),A^2 -> H(U1(k, d), U1(e, d))) (A^2 -> H(U1(c, k), U1(e, d)),A^2 -> H(U1(c, k), U1(e, d))) (A^2 -> H(U1(c, d), d),A^2 -> H(U1(c, d), d)) (A^2 -> H(U1(c, d), U1(e, k)),A^2 -> H(U1(c, d), U1(e, k))) ---------------------------------------- (1904) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), U1(d, e)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1905) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(d, e)),A^2 -> H(U1(e, d), U1(d, e))) (A^2 -> H(U1(k, d), U1(d, e)),A^2 -> H(U1(k, d), U1(d, e))) (A^2 -> H(U1(c, k), U1(d, e)),A^2 -> H(U1(c, k), U1(d, e))) (A^2 -> H(U1(c, d), U1(k, e)),A^2 -> H(U1(c, d), U1(k, e))) ---------------------------------------- (1906) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), c) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1907) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), c),A^2 -> H(U1(e, d), c)) (A^2 -> H(U1(k, d), c),A^2 -> H(U1(k, d), c)) (A^2 -> H(U1(c, k), c),A^2 -> H(U1(c, k), c)) (A^2 -> H(U1(c, d), e),A^2 -> H(U1(c, d), e)) (A^2 -> H(U1(c, d), k),A^2 -> H(U1(c, d), k)) ---------------------------------------- (1908) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), U1(e, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1909) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(e, k)),A^2 -> H(U1(e, d), U1(e, k))) (A^2 -> H(U1(k, d), U1(e, k)),A^2 -> H(U1(k, d), U1(e, k))) (A^2 -> H(U1(c, k), U1(e, k)),A^2 -> H(U1(c, k), U1(e, k))) (A^2 -> H(U1(c, d), k),A^2 -> H(U1(c, d), k)) ---------------------------------------- (1910) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), U1(k, e)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1911) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(k, e)),A^2 -> H(U1(e, d), U1(k, e))) (A^2 -> H(U1(k, d), U1(k, e)),A^2 -> H(U1(k, d), U1(k, e))) (A^2 -> H(U1(c, k), U1(k, e)),A^2 -> H(U1(c, k), U1(k, e))) ---------------------------------------- (1912) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), e) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1913) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), e),A^2 -> H(U1(e, d), e)) (A^2 -> H(U1(k, d), e),A^2 -> H(U1(k, d), e)) (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) ---------------------------------------- (1914) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(k, d), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1915) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1916) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, d), U1(k, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1917) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), U1(k, k)),A^2 -> H(U1(e, d), U1(k, k))) (A^2 -> H(U1(k, d), U1(k, k)),A^2 -> H(U1(k, d), U1(k, k))) (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) ---------------------------------------- (1918) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, a), d) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1919) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), d),A^2 -> H(U1(e, a), d)) (A^2 -> H(U1(k, a), d),A^2 -> H(U1(k, a), d)) (A^2 -> H(U1(c, c), d),A^2 -> H(U1(c, c), d)) (A^2 -> H(U1(c, d), d),A^2 -> H(U1(c, d), d)) (A^2 -> H(U1(c, a), k),A^2 -> H(U1(c, a), k)) ---------------------------------------- (1920) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, a), k) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1921) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, a), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, a), k),A^2 -> H(U1(e, a), k)) (A^2 -> H(U1(k, a), k),A^2 -> H(U1(k, a), k)) (A^2 -> H(U1(c, c), k),A^2 -> H(U1(c, c), k)) (A^2 -> H(U1(c, d), k),A^2 -> H(U1(c, d), k)) ---------------------------------------- (1922) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(k, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1923) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1924) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(e, b)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1925) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, b)),A^2 -> H(U1(k, e), U1(e, b))) (A^2 -> H(U1(d, e), b),A^2 -> H(U1(d, e), b)) (A^2 -> H(U1(d, e), U1(e, c)),A^2 -> H(U1(d, e), U1(e, c))) (A^2 -> H(U1(d, e), U1(e, d)),A^2 -> H(U1(d, e), U1(e, d))) ---------------------------------------- (1926) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(k, b)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1927) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, b)),A^2 -> H(U1(k, e), U1(k, b))) (A^2 -> H(U1(d, e), U1(k, c)),A^2 -> H(U1(d, e), U1(k, c))) (A^2 -> H(U1(d, e), U1(k, d)),A^2 -> H(U1(d, e), U1(k, d))) ---------------------------------------- (1928) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(c, d)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1929) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, d)),A^2 -> H(U1(k, e), U1(c, d))) (A^2 -> H(U1(d, e), U1(e, d)),A^2 -> H(U1(d, e), U1(e, d))) (A^2 -> H(U1(d, e), U1(k, d)),A^2 -> H(U1(d, e), U1(k, d))) (A^2 -> H(U1(d, e), U1(c, k)),A^2 -> H(U1(d, e), U1(c, k))) ---------------------------------------- (1930) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(d, c)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1931) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, c)),A^2 -> H(U1(k, e), U1(d, c))) (A^2 -> H(U1(d, e), U1(k, c)),A^2 -> H(U1(d, e), U1(k, c))) (A^2 -> H(U1(d, e), U1(d, e)),A^2 -> H(U1(d, e), U1(d, e))) (A^2 -> H(U1(d, e), U1(d, k)),A^2 -> H(U1(d, e), U1(d, k))) ---------------------------------------- (1932) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(b, e)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1933) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, e)),A^2 -> H(U1(k, e), U1(b, e))) (A^2 -> H(U1(d, e), U1(c, e)),A^2 -> H(U1(d, e), U1(c, e))) (A^2 -> H(U1(d, e), U1(d, e)),A^2 -> H(U1(d, e), U1(d, e))) ---------------------------------------- (1934) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(b, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1935) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(b, k)),A^2 -> H(U1(k, e), U1(b, k))) (A^2 -> H(U1(d, e), U1(c, k)),A^2 -> H(U1(d, e), U1(c, k))) (A^2 -> H(U1(d, e), U1(d, k)),A^2 -> H(U1(d, e), U1(d, k))) ---------------------------------------- (1936) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(e, c)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1937) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, c)),A^2 -> H(U1(k, e), U1(e, c))) (A^2 -> H(U1(d, e), c),A^2 -> H(U1(d, e), c)) (A^2 -> H(U1(d, e), U1(e, e)),A^2 -> H(U1(d, e), U1(e, e))) (A^2 -> H(U1(d, e), U1(e, k)),A^2 -> H(U1(d, e), U1(e, k))) ---------------------------------------- (1938) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(k, c)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1939) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, c)),A^2 -> H(U1(k, e), U1(k, c))) (A^2 -> H(U1(d, e), U1(k, e)),A^2 -> H(U1(d, e), U1(k, e))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1940) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(c, e)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1941) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, e)),A^2 -> H(U1(k, e), U1(c, e))) (A^2 -> H(U1(d, e), U1(e, e)),A^2 -> H(U1(d, e), U1(e, e))) (A^2 -> H(U1(d, e), U1(k, e)),A^2 -> H(U1(d, e), U1(k, e))) ---------------------------------------- (1942) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(c, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1943) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, k)),A^2 -> H(U1(k, e), U1(c, k))) (A^2 -> H(U1(d, e), U1(e, k)),A^2 -> H(U1(d, e), U1(e, k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1944) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(e, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1945) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(d, e), e),A^2 -> H(U1(d, e), e)) ---------------------------------------- (1946) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1947) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (1948) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1949) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1950) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(k, d)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1951) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, d)),A^2 -> H(U1(k, e), U1(k, d))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1952) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, e), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1953) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, k)),A^2 -> H(U1(k, e), U1(d, k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1954) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), b) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1955) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), b),A^2 -> H(U1(k, c), b)) (A^2 -> H(U1(d, e), b),A^2 -> H(U1(d, e), b)) (A^2 -> H(U1(d, k), b),A^2 -> H(U1(d, k), b)) (A^2 -> H(U1(d, c), c),A^2 -> H(U1(d, c), c)) (A^2 -> H(U1(d, c), d),A^2 -> H(U1(d, c), d)) ---------------------------------------- (1956) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), U1(e, d)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1957) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, d)),A^2 -> H(U1(k, c), U1(e, d))) (A^2 -> H(U1(d, e), U1(e, d)),A^2 -> H(U1(d, e), U1(e, d))) (A^2 -> H(U1(d, k), U1(e, d)),A^2 -> H(U1(d, k), U1(e, d))) (A^2 -> H(U1(d, c), d),A^2 -> H(U1(d, c), d)) (A^2 -> H(U1(d, c), U1(e, k)),A^2 -> H(U1(d, c), U1(e, k))) ---------------------------------------- (1958) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), U1(d, e)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1959) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(d, e)),A^2 -> H(U1(k, c), U1(d, e))) (A^2 -> H(U1(d, e), U1(d, e)),A^2 -> H(U1(d, e), U1(d, e))) (A^2 -> H(U1(d, k), U1(d, e)),A^2 -> H(U1(d, k), U1(d, e))) (A^2 -> H(U1(d, c), U1(k, e)),A^2 -> H(U1(d, c), U1(k, e))) ---------------------------------------- (1960) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), c) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1961) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), c),A^2 -> H(U1(k, c), c)) (A^2 -> H(U1(d, e), c),A^2 -> H(U1(d, e), c)) (A^2 -> H(U1(d, k), c),A^2 -> H(U1(d, k), c)) (A^2 -> H(U1(d, c), e),A^2 -> H(U1(d, c), e)) (A^2 -> H(U1(d, c), k),A^2 -> H(U1(d, c), k)) ---------------------------------------- (1962) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), U1(e, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1963) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(e, k)),A^2 -> H(U1(k, c), U1(e, k))) (A^2 -> H(U1(d, e), U1(e, k)),A^2 -> H(U1(d, e), U1(e, k))) (A^2 -> H(U1(d, k), U1(e, k)),A^2 -> H(U1(d, k), U1(e, k))) (A^2 -> H(U1(d, c), k),A^2 -> H(U1(d, c), k)) ---------------------------------------- (1964) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), U1(k, e)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1965) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, e)),A^2 -> H(U1(k, c), U1(k, e))) (A^2 -> H(U1(d, e), U1(k, e)),A^2 -> H(U1(d, e), U1(k, e))) (A^2 -> H(U1(d, k), U1(k, e)),A^2 -> H(U1(d, k), U1(k, e))) ---------------------------------------- (1966) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), e) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1967) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), e),A^2 -> H(U1(k, c), e)) (A^2 -> H(U1(d, e), e),A^2 -> H(U1(d, e), e)) (A^2 -> H(U1(d, k), e),A^2 -> H(U1(d, k), e)) ---------------------------------------- (1968) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(k, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1969) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1970) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, c), U1(k, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1971) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), U1(k, k)),A^2 -> H(U1(k, c), U1(k, k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (1972) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, a), d) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1973) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), d),A^2 -> H(U1(k, a), d)) (A^2 -> H(U1(d, c), d),A^2 -> H(U1(d, c), d)) (A^2 -> H(U1(d, d), d),A^2 -> H(U1(d, d), d)) (A^2 -> H(U1(d, a), k),A^2 -> H(U1(d, a), k)) ---------------------------------------- (1974) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, a), k) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1975) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, a), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, a), k),A^2 -> H(U1(k, a), k)) (A^2 -> H(U1(d, c), k),A^2 -> H(U1(d, c), k)) (A^2 -> H(U1(d, d), k),A^2 -> H(U1(d, d), k)) ---------------------------------------- (1976) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(k, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1977) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (1978) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), b) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1979) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), b),A^2 -> H(U1(c, e), b)) (A^2 -> H(U1(d, e), b),A^2 -> H(U1(d, e), b)) (A^2 -> H(U1(a, e), c),A^2 -> H(U1(a, e), c)) (A^2 -> H(U1(a, e), d),A^2 -> H(U1(a, e), d)) ---------------------------------------- (1980) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), U1(e, d)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1981) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(e, d)),A^2 -> H(U1(c, e), U1(e, d))) (A^2 -> H(U1(d, e), U1(e, d)),A^2 -> H(U1(d, e), U1(e, d))) (A^2 -> H(U1(a, e), d),A^2 -> H(U1(a, e), d)) (A^2 -> H(U1(a, e), U1(e, k)),A^2 -> H(U1(a, e), U1(e, k))) ---------------------------------------- (1982) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), U1(d, e)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1983) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(d, e)),A^2 -> H(U1(c, e), U1(d, e))) (A^2 -> H(U1(d, e), U1(d, e)),A^2 -> H(U1(d, e), U1(d, e))) (A^2 -> H(U1(a, e), U1(k, e)),A^2 -> H(U1(a, e), U1(k, e))) ---------------------------------------- (1984) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), c) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1985) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), c),A^2 -> H(U1(c, e), c)) (A^2 -> H(U1(d, e), c),A^2 -> H(U1(d, e), c)) (A^2 -> H(U1(a, e), e),A^2 -> H(U1(a, e), e)) (A^2 -> H(U1(a, e), k),A^2 -> H(U1(a, e), k)) ---------------------------------------- (1986) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), U1(e, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1987) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(e, k)),A^2 -> H(U1(c, e), U1(e, k))) (A^2 -> H(U1(d, e), U1(e, k)),A^2 -> H(U1(d, e), U1(e, k))) (A^2 -> H(U1(a, e), k),A^2 -> H(U1(a, e), k)) ---------------------------------------- (1988) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), U1(k, e)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1989) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(k, e)),A^2 -> H(U1(c, e), U1(k, e))) (A^2 -> H(U1(d, e), U1(k, e)),A^2 -> H(U1(d, e), U1(k, e))) ---------------------------------------- (1990) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), e) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1991) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), e),A^2 -> H(U1(c, e), e)) (A^2 -> H(U1(d, e), e),A^2 -> H(U1(d, e), e)) ---------------------------------------- (1992) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, e), U1(k, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1993) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), U1(k, k)),A^2 -> H(U1(c, e), U1(k, k))) (A^2 -> H(U1(d, e), U1(k, k)),A^2 -> H(U1(d, e), U1(k, k))) ---------------------------------------- (1994) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), b) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1995) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), b),A^2 -> H(U1(c, k), b)) (A^2 -> H(U1(d, k), b),A^2 -> H(U1(d, k), b)) (A^2 -> H(U1(a, k), c),A^2 -> H(U1(a, k), c)) (A^2 -> H(U1(a, k), d),A^2 -> H(U1(a, k), d)) ---------------------------------------- (1996) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), U1(e, d)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1997) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(e, d)),A^2 -> H(U1(c, k), U1(e, d))) (A^2 -> H(U1(d, k), U1(e, d)),A^2 -> H(U1(d, k), U1(e, d))) (A^2 -> H(U1(a, k), d),A^2 -> H(U1(a, k), d)) (A^2 -> H(U1(a, k), U1(e, k)),A^2 -> H(U1(a, k), U1(e, k))) ---------------------------------------- (1998) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), U1(d, e)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (1999) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(d, e)),A^2 -> H(U1(c, k), U1(d, e))) (A^2 -> H(U1(d, k), U1(d, e)),A^2 -> H(U1(d, k), U1(d, e))) (A^2 -> H(U1(a, k), U1(k, e)),A^2 -> H(U1(a, k), U1(k, e))) ---------------------------------------- (2000) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), c) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2001) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), c),A^2 -> H(U1(c, k), c)) (A^2 -> H(U1(d, k), c),A^2 -> H(U1(d, k), c)) (A^2 -> H(U1(a, k), e),A^2 -> H(U1(a, k), e)) (A^2 -> H(U1(a, k), k),A^2 -> H(U1(a, k), k)) ---------------------------------------- (2002) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), U1(e, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2003) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(e, k)),A^2 -> H(U1(c, k), U1(e, k))) (A^2 -> H(U1(d, k), U1(e, k)),A^2 -> H(U1(d, k), U1(e, k))) (A^2 -> H(U1(a, k), k),A^2 -> H(U1(a, k), k)) ---------------------------------------- (2004) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), U1(k, e)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2005) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(k, e)),A^2 -> H(U1(c, k), U1(k, e))) (A^2 -> H(U1(d, k), U1(k, e)),A^2 -> H(U1(d, k), U1(k, e))) ---------------------------------------- (2006) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), e) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2007) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) (A^2 -> H(U1(d, k), e),A^2 -> H(U1(d, k), e)) ---------------------------------------- (2008) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, k), U1(k, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2009) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), U1(k, k)),A^2 -> H(U1(c, k), U1(k, k))) (A^2 -> H(U1(d, k), U1(k, k)),A^2 -> H(U1(d, k), U1(k, k))) ---------------------------------------- (2010) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, c), d) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2011) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), d),A^2 -> H(U1(c, c), d)) (A^2 -> H(U1(d, c), d),A^2 -> H(U1(d, c), d)) (A^2 -> H(U1(a, e), d),A^2 -> H(U1(a, e), d)) (A^2 -> H(U1(a, k), d),A^2 -> H(U1(a, k), d)) (A^2 -> H(U1(a, c), k),A^2 -> H(U1(a, c), k)) ---------------------------------------- (2012) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, c), k) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2013) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, c), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, c), k),A^2 -> H(U1(c, c), k)) (A^2 -> H(U1(d, c), k),A^2 -> H(U1(d, c), k)) (A^2 -> H(U1(a, e), k),A^2 -> H(U1(a, e), k)) (A^2 -> H(U1(a, k), k),A^2 -> H(U1(a, k), k)) ---------------------------------------- (2014) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, d), d) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2015) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), d),A^2 -> H(U1(c, d), d)) (A^2 -> H(U1(d, d), d),A^2 -> H(U1(d, d), d)) (A^2 -> H(U1(a, k), d),A^2 -> H(U1(a, k), d)) (A^2 -> H(U1(a, d), k),A^2 -> H(U1(a, d), k)) ---------------------------------------- (2016) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(a, d), k) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2017) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, d), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, d), k),A^2 -> H(U1(c, d), k)) (A^2 -> H(U1(d, d), k),A^2 -> H(U1(d, d), k)) (A^2 -> H(U1(a, k), k),A^2 -> H(U1(a, k), k)) ---------------------------------------- (2018) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(c, b)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2019) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(c, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, b)),A^2 -> H(k, U1(e, b))) (A^2 -> H(k, U1(k, b)),A^2 -> H(k, U1(k, b))) (A^2 -> H(k, U1(c, c)),A^2 -> H(k, U1(c, c))) (A^2 -> H(k, U1(c, d)),A^2 -> H(k, U1(c, d))) ---------------------------------------- (2020) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(k, b)) A^2 -> H(k, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2021) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2022) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(d, b)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2023) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(d, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, b)),A^2 -> H(k, U1(k, b))) (A^2 -> H(k, U1(d, c)),A^2 -> H(k, U1(d, c))) (A^2 -> H(k, U1(d, d)),A^2 -> H(k, U1(d, d))) ---------------------------------------- (2024) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(k, b)) A^2 -> H(k, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2025) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2026) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(b, c)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2027) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(b, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, c)),A^2 -> H(k, U1(c, c))) (A^2 -> H(k, U1(d, c)),A^2 -> H(k, U1(d, c))) (A^2 -> H(k, U1(b, e)),A^2 -> H(k, U1(b, e))) (A^2 -> H(k, U1(b, k)),A^2 -> H(k, U1(b, k))) ---------------------------------------- (2028) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(b, d)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2029) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(b, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, d)),A^2 -> H(k, U1(c, d))) (A^2 -> H(k, U1(d, d)),A^2 -> H(k, U1(d, d))) (A^2 -> H(k, U1(b, k)),A^2 -> H(k, U1(b, k))) ---------------------------------------- (2030) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(c, c)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2031) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(c, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, c)),A^2 -> H(k, U1(e, c))) (A^2 -> H(k, U1(k, c)),A^2 -> H(k, U1(k, c))) (A^2 -> H(k, U1(c, e)),A^2 -> H(k, U1(c, e))) (A^2 -> H(k, U1(c, k)),A^2 -> H(k, U1(c, k))) ---------------------------------------- (2032) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(k, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2033) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2034) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, f(e)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2035) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, f(e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) ---------------------------------------- (2036) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, f(k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2037) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, f(k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2038) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2039) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2040) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(k, U1(d, d)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2041) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(d, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, d)),A^2 -> H(k, U1(k, d))) (A^2 -> H(k, U1(d, k)),A^2 -> H(k, U1(d, k))) ---------------------------------------- (2042) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(k, d)) A^2 -> H(k, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2043) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2044) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(e, b)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2045) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, b)),A^2 -> H(e, U1(e, b))) (A^2 -> H(k, U1(e, b)),A^2 -> H(k, U1(e, b))) (A^2 -> H(c, b),A^2 -> H(c, b)) (A^2 -> H(c, U1(e, c)),A^2 -> H(c, U1(e, c))) (A^2 -> H(c, U1(e, d)),A^2 -> H(c, U1(e, d))) ---------------------------------------- (2046) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(k, b)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2047) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, b)),A^2 -> H(e, U1(k, b))) (A^2 -> H(k, U1(k, b)),A^2 -> H(k, U1(k, b))) (A^2 -> H(c, U1(k, c)),A^2 -> H(c, U1(k, c))) (A^2 -> H(c, U1(k, d)),A^2 -> H(c, U1(k, d))) ---------------------------------------- (2048) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(e, U1(k, b)) A^2 -> H(k, U1(k, b)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2049) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2050) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(c, d)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2051) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, d)),A^2 -> H(e, U1(c, d))) (A^2 -> H(k, U1(c, d)),A^2 -> H(k, U1(c, d))) (A^2 -> H(c, U1(e, d)),A^2 -> H(c, U1(e, d))) (A^2 -> H(c, U1(k, d)),A^2 -> H(c, U1(k, d))) (A^2 -> H(c, U1(c, k)),A^2 -> H(c, U1(c, k))) ---------------------------------------- (2052) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(d, c)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2053) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, c)),A^2 -> H(e, U1(d, c))) (A^2 -> H(k, U1(d, c)),A^2 -> H(k, U1(d, c))) (A^2 -> H(c, U1(k, c)),A^2 -> H(c, U1(k, c))) (A^2 -> H(c, U1(d, e)),A^2 -> H(c, U1(d, e))) (A^2 -> H(c, U1(d, k)),A^2 -> H(c, U1(d, k))) ---------------------------------------- (2054) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(b, e)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2055) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, e)),A^2 -> H(e, U1(b, e))) (A^2 -> H(k, U1(b, e)),A^2 -> H(k, U1(b, e))) (A^2 -> H(c, U1(c, e)),A^2 -> H(c, U1(c, e))) (A^2 -> H(c, U1(d, e)),A^2 -> H(c, U1(d, e))) ---------------------------------------- (2056) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(b, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2057) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(b, k)),A^2 -> H(e, U1(b, k))) (A^2 -> H(k, U1(b, k)),A^2 -> H(k, U1(b, k))) (A^2 -> H(c, U1(c, k)),A^2 -> H(c, U1(c, k))) (A^2 -> H(c, U1(d, k)),A^2 -> H(c, U1(d, k))) ---------------------------------------- (2058) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(e, c)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2059) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, c)),A^2 -> H(e, U1(e, c))) (A^2 -> H(k, U1(e, c)),A^2 -> H(k, U1(e, c))) (A^2 -> H(c, c),A^2 -> H(c, c)) (A^2 -> H(c, U1(e, e)),A^2 -> H(c, U1(e, e))) (A^2 -> H(c, U1(e, k)),A^2 -> H(c, U1(e, k))) ---------------------------------------- (2060) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(k, c)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2061) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, c)),A^2 -> H(e, U1(k, c))) (A^2 -> H(k, U1(k, c)),A^2 -> H(k, U1(k, c))) (A^2 -> H(c, U1(k, e)),A^2 -> H(c, U1(k, e))) (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) ---------------------------------------- (2062) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(e, U1(k, c)) A^2 -> H(k, U1(k, c)) A^2 -> H(c, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2063) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2064) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(c, e)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2065) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, e)),A^2 -> H(e, U1(c, e))) (A^2 -> H(k, U1(c, e)),A^2 -> H(k, U1(c, e))) (A^2 -> H(c, U1(e, e)),A^2 -> H(c, U1(e, e))) (A^2 -> H(c, U1(k, e)),A^2 -> H(c, U1(k, e))) ---------------------------------------- (2066) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(c, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2067) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, k)),A^2 -> H(e, U1(c, k))) (A^2 -> H(k, U1(c, k)),A^2 -> H(k, U1(c, k))) (A^2 -> H(c, U1(e, k)),A^2 -> H(c, U1(e, k))) (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) ---------------------------------------- (2068) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(e, e)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2069) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, e)),A^2 -> H(e, U1(e, e))) (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(c, e),A^2 -> H(c, e)) ---------------------------------------- (2070) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(k, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2071) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2072) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(e, U1(k, k)) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2073) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2074) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(k, d)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2075) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, d)),A^2 -> H(e, U1(k, d))) (A^2 -> H(k, U1(k, d)),A^2 -> H(k, U1(k, d))) (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) ---------------------------------------- (2076) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(e, U1(k, d)) A^2 -> H(k, U1(k, d)) A^2 -> H(c, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2077) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2078) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(c, U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2079) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, k)),A^2 -> H(e, U1(d, k))) (A^2 -> H(k, U1(d, k)),A^2 -> H(k, U1(d, k))) (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) ---------------------------------------- (2080) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(e, b)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2081) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, b)),A^2 -> H(k, U1(e, b))) (A^2 -> H(U1(e, k), b),A^2 -> H(U1(e, k), b)) (A^2 -> H(U1(e, k), U1(e, c)),A^2 -> H(U1(e, k), U1(e, c))) (A^2 -> H(U1(e, k), U1(e, d)),A^2 -> H(U1(e, k), U1(e, d))) ---------------------------------------- (2082) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(k, b)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2083) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, b)),A^2 -> H(k, U1(k, b))) (A^2 -> H(U1(e, k), U1(k, c)),A^2 -> H(U1(e, k), U1(k, c))) (A^2 -> H(U1(e, k), U1(k, d)),A^2 -> H(U1(e, k), U1(k, d))) ---------------------------------------- (2084) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(k, U1(k, b)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2085) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2086) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(c, d)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2087) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, d)),A^2 -> H(k, U1(c, d))) (A^2 -> H(U1(e, k), U1(e, d)),A^2 -> H(U1(e, k), U1(e, d))) (A^2 -> H(U1(e, k), U1(k, d)),A^2 -> H(U1(e, k), U1(k, d))) (A^2 -> H(U1(e, k), U1(c, k)),A^2 -> H(U1(e, k), U1(c, k))) ---------------------------------------- (2088) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(d, c)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2089) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, c)),A^2 -> H(k, U1(d, c))) (A^2 -> H(U1(e, k), U1(k, c)),A^2 -> H(U1(e, k), U1(k, c))) (A^2 -> H(U1(e, k), U1(d, e)),A^2 -> H(U1(e, k), U1(d, e))) (A^2 -> H(U1(e, k), U1(d, k)),A^2 -> H(U1(e, k), U1(d, k))) ---------------------------------------- (2090) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(b, e)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2091) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, e)),A^2 -> H(k, U1(b, e))) (A^2 -> H(U1(e, k), U1(c, e)),A^2 -> H(U1(e, k), U1(c, e))) (A^2 -> H(U1(e, k), U1(d, e)),A^2 -> H(U1(e, k), U1(d, e))) ---------------------------------------- (2092) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(b, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2093) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, k)),A^2 -> H(k, U1(b, k))) (A^2 -> H(U1(e, k), U1(c, k)),A^2 -> H(U1(e, k), U1(c, k))) (A^2 -> H(U1(e, k), U1(d, k)),A^2 -> H(U1(e, k), U1(d, k))) ---------------------------------------- (2094) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(e, c)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2095) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, c)),A^2 -> H(k, U1(e, c))) (A^2 -> H(U1(e, k), c),A^2 -> H(U1(e, k), c)) (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(e, k), U1(e, k)),A^2 -> H(U1(e, k), U1(e, k))) ---------------------------------------- (2096) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(k, c)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2097) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, c)),A^2 -> H(k, U1(k, c))) (A^2 -> H(U1(e, k), U1(k, e)),A^2 -> H(U1(e, k), U1(k, e))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (2098) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(k, U1(k, c)) A^2 -> H(U1(e, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2099) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2100) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(c, e)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2101) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, e)),A^2 -> H(k, U1(c, e))) (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(e, k), U1(k, e)),A^2 -> H(U1(e, k), U1(k, e))) ---------------------------------------- (2102) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(c, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2103) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, k)),A^2 -> H(k, U1(c, k))) (A^2 -> H(U1(e, k), U1(e, k)),A^2 -> H(U1(e, k), U1(e, k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (2104) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2105) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) ---------------------------------------- (2106) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2107) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2108) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2109) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2110) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(k, d)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2111) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, d)),A^2 -> H(k, U1(k, d))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (2112) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(k, U1(k, d)) A^2 -> H(U1(e, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2113) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2114) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2115) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, k)),A^2 -> H(k, U1(d, k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (2116) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), b) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2117) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, b),A^2 -> H(c, b)) (A^2 -> H(U1(e, e), b),A^2 -> H(U1(e, e), b)) (A^2 -> H(U1(e, k), b),A^2 -> H(U1(e, k), b)) (A^2 -> H(U1(e, c), c),A^2 -> H(U1(e, c), c)) (A^2 -> H(U1(e, c), d),A^2 -> H(U1(e, c), d)) ---------------------------------------- (2118) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(e, d)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2119) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, d)),A^2 -> H(c, U1(e, d))) (A^2 -> H(U1(e, e), U1(e, d)),A^2 -> H(U1(e, e), U1(e, d))) (A^2 -> H(U1(e, k), U1(e, d)),A^2 -> H(U1(e, k), U1(e, d))) (A^2 -> H(U1(e, c), d),A^2 -> H(U1(e, c), d)) (A^2 -> H(U1(e, c), U1(e, k)),A^2 -> H(U1(e, c), U1(e, k))) ---------------------------------------- (2120) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(d, e)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2121) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, e)),A^2 -> H(c, U1(d, e))) (A^2 -> H(U1(e, e), U1(d, e)),A^2 -> H(U1(e, e), U1(d, e))) (A^2 -> H(U1(e, k), U1(d, e)),A^2 -> H(U1(e, k), U1(d, e))) (A^2 -> H(U1(e, c), U1(k, e)),A^2 -> H(U1(e, c), U1(k, e))) ---------------------------------------- (2122) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(k, e)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2123) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, e)),A^2 -> H(c, U1(k, e))) (A^2 -> H(U1(e, e), U1(k, e)),A^2 -> H(U1(e, e), U1(k, e))) (A^2 -> H(U1(e, k), U1(k, e)),A^2 -> H(U1(e, k), U1(k, e))) ---------------------------------------- (2124) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(e, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2125) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, k)),A^2 -> H(c, U1(e, k))) (A^2 -> H(U1(e, e), U1(e, k)),A^2 -> H(U1(e, e), U1(e, k))) (A^2 -> H(U1(e, k), U1(e, k)),A^2 -> H(U1(e, k), U1(e, k))) (A^2 -> H(U1(e, c), k),A^2 -> H(U1(e, c), k)) ---------------------------------------- (2126) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), e) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2127) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, e),A^2 -> H(c, e)) (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) ---------------------------------------- (2128) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(k, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2129) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (2130) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(e, b)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2131) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), b),A^2 -> H(U1(k, e), b)) (A^2 -> H(U1(k, e), U1(e, c)),A^2 -> H(U1(k, e), U1(e, c))) (A^2 -> H(U1(k, e), U1(e, d)),A^2 -> H(U1(k, e), U1(e, d))) ---------------------------------------- (2132) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(k, b)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2133) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, c)),A^2 -> H(U1(k, e), U1(k, c))) (A^2 -> H(U1(k, e), U1(k, d)),A^2 -> H(U1(k, e), U1(k, d))) ---------------------------------------- (2134) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(c, d)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2135) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, d)),A^2 -> H(U1(k, e), U1(e, d))) (A^2 -> H(U1(k, e), U1(k, d)),A^2 -> H(U1(k, e), U1(k, d))) (A^2 -> H(U1(k, e), U1(c, k)),A^2 -> H(U1(k, e), U1(c, k))) ---------------------------------------- (2136) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(d, c)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2137) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, c)),A^2 -> H(U1(k, e), U1(k, c))) (A^2 -> H(U1(k, e), U1(d, e)),A^2 -> H(U1(k, e), U1(d, e))) (A^2 -> H(U1(k, e), U1(d, k)),A^2 -> H(U1(k, e), U1(d, k))) ---------------------------------------- (2138) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(b, e)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2139) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, e)),A^2 -> H(U1(k, e), U1(c, e))) (A^2 -> H(U1(k, e), U1(d, e)),A^2 -> H(U1(k, e), U1(d, e))) ---------------------------------------- (2140) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(b, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2141) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(c, k)),A^2 -> H(U1(k, e), U1(c, k))) (A^2 -> H(U1(k, e), U1(d, k)),A^2 -> H(U1(k, e), U1(d, k))) ---------------------------------------- (2142) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(e, c)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2143) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), c),A^2 -> H(U1(k, e), c)) (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(k, e), U1(e, k)),A^2 -> H(U1(k, e), U1(e, k))) ---------------------------------------- (2144) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(k, c)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2145) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, e)),A^2 -> H(U1(k, e), U1(k, e))) (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (2146) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2147) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2148) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(c, e)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2149) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(k, e), U1(k, e)),A^2 -> H(U1(k, e), U1(k, e))) ---------------------------------------- (2150) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(c, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2151) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, k)),A^2 -> H(U1(k, e), U1(e, k))) (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (2152) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2153) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2154) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2155) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), e),A^2 -> H(U1(k, e), e)) ---------------------------------------- (2156) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2157) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2158) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(k, d)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2159) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (2160) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2161) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2162) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, e), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2163) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (2164) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2165) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2166) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), b) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2167) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), b),A^2 -> H(U1(k, e), b)) (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(U1(k, c), c),A^2 -> H(U1(k, c), c)) (A^2 -> H(U1(k, c), d),A^2 -> H(U1(k, c), d)) ---------------------------------------- (2168) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(e, d)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2169) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, d)),A^2 -> H(U1(k, e), U1(e, d))) (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(U1(k, c), d),A^2 -> H(U1(k, c), d)) (A^2 -> H(U1(k, c), U1(e, k)),A^2 -> H(U1(k, c), U1(e, k))) ---------------------------------------- (2170) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(d, e)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2171) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(d, e)),A^2 -> H(U1(k, e), U1(d, e))) (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(U1(k, c), U1(k, e)),A^2 -> H(U1(k, c), U1(k, e))) ---------------------------------------- (2172) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), c) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2173) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), c),A^2 -> H(U1(k, e), c)) (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(k, c), e),A^2 -> H(U1(k, c), e)) (A^2 -> H(U1(k, c), k),A^2 -> H(U1(k, c), k)) ---------------------------------------- (2174) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(k, c), e) A^2 -> H(U1(k, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2175) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2176) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(e, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2177) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, k)),A^2 -> H(U1(k, e), U1(e, k))) (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(U1(k, c), k),A^2 -> H(U1(k, c), k)) ---------------------------------------- (2178) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(k, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2179) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2180) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(k, e)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2181) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, e)),A^2 -> H(U1(k, e), U1(k, e))) (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2182) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2183) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2184) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(k, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2185) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2186) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2187) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2188) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), b) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2189) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), b),A^2 -> H(U1(e, e), b)) (A^2 -> H(U1(k, e), b),A^2 -> H(U1(k, e), b)) (A^2 -> H(U1(c, e), c),A^2 -> H(U1(c, e), c)) (A^2 -> H(U1(c, e), d),A^2 -> H(U1(c, e), d)) ---------------------------------------- (2190) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(e, d)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2191) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, d)),A^2 -> H(U1(e, e), U1(e, d))) (A^2 -> H(U1(k, e), U1(e, d)),A^2 -> H(U1(k, e), U1(e, d))) (A^2 -> H(U1(c, e), d),A^2 -> H(U1(c, e), d)) (A^2 -> H(U1(c, e), U1(e, k)),A^2 -> H(U1(c, e), U1(e, k))) ---------------------------------------- (2192) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(d, e)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2193) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(d, e)),A^2 -> H(U1(e, e), U1(d, e))) (A^2 -> H(U1(k, e), U1(d, e)),A^2 -> H(U1(k, e), U1(d, e))) (A^2 -> H(U1(c, e), U1(k, e)),A^2 -> H(U1(c, e), U1(k, e))) ---------------------------------------- (2194) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), c) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2195) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), c),A^2 -> H(U1(e, e), c)) (A^2 -> H(U1(k, e), c),A^2 -> H(U1(k, e), c)) (A^2 -> H(U1(c, e), e),A^2 -> H(U1(c, e), e)) (A^2 -> H(U1(c, e), k),A^2 -> H(U1(c, e), k)) ---------------------------------------- (2196) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(e, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2197) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, k)),A^2 -> H(U1(e, e), U1(e, k))) (A^2 -> H(U1(k, e), U1(e, k)),A^2 -> H(U1(k, e), U1(e, k))) (A^2 -> H(U1(c, e), k),A^2 -> H(U1(c, e), k)) ---------------------------------------- (2198) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(k, e)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2199) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, e)),A^2 -> H(U1(e, e), U1(k, e))) (A^2 -> H(U1(k, e), U1(k, e)),A^2 -> H(U1(k, e), U1(k, e))) ---------------------------------------- (2200) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2201) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) (A^2 -> H(U1(k, e), e),A^2 -> H(U1(k, e), e)) ---------------------------------------- (2202) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(k, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2203) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2204) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(k, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2205) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(k, k)),A^2 -> H(U1(e, e), U1(k, k))) (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (2206) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2207) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2208) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), b) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2209) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), b),A^2 -> H(U1(e, k), b)) (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(U1(c, k), c),A^2 -> H(U1(c, k), c)) (A^2 -> H(U1(c, k), d),A^2 -> H(U1(c, k), d)) ---------------------------------------- (2210) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(e, d)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2211) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(e, d)),A^2 -> H(U1(e, k), U1(e, d))) (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(U1(c, k), d),A^2 -> H(U1(c, k), d)) (A^2 -> H(U1(c, k), U1(e, k)),A^2 -> H(U1(c, k), U1(e, k))) ---------------------------------------- (2212) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(d, e)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2213) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(d, e)),A^2 -> H(U1(e, k), U1(d, e))) (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(U1(c, k), U1(k, e)),A^2 -> H(U1(c, k), U1(k, e))) ---------------------------------------- (2214) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), c) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2215) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), c),A^2 -> H(U1(e, k), c)) (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) (A^2 -> H(U1(c, k), k),A^2 -> H(U1(c, k), k)) ---------------------------------------- (2216) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(e, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2217) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(e, k)),A^2 -> H(U1(e, k), U1(e, k))) (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(U1(c, k), k),A^2 -> H(U1(c, k), k)) ---------------------------------------- (2218) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(k, e)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2219) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(k, e)),A^2 -> H(U1(e, k), U1(k, e))) (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2220) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2221) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2222) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), e) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2223) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (2224) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2225) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2226) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(k, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2227) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2228) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), d) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2229) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), d),A^2 -> H(U1(e, c), d)) (A^2 -> H(U1(k, c), d),A^2 -> H(U1(k, c), d)) (A^2 -> H(U1(c, e), d),A^2 -> H(U1(c, e), d)) (A^2 -> H(U1(c, k), d),A^2 -> H(U1(c, k), d)) (A^2 -> H(U1(c, c), k),A^2 -> H(U1(c, c), k)) ---------------------------------------- (2230) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(e, b)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2231) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, b),A^2 -> H(e, b)) (A^2 -> H(e, U1(e, c)),A^2 -> H(e, U1(e, c))) (A^2 -> H(e, U1(e, d)),A^2 -> H(e, U1(e, d))) ---------------------------------------- (2232) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(c, d)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2233) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, d)),A^2 -> H(e, U1(e, d))) (A^2 -> H(e, U1(k, d)),A^2 -> H(e, U1(k, d))) (A^2 -> H(e, U1(c, k)),A^2 -> H(e, U1(c, k))) ---------------------------------------- (2234) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(k, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2235) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2236) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(d, c)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2237) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, c)),A^2 -> H(e, U1(k, c))) (A^2 -> H(e, U1(d, e)),A^2 -> H(e, U1(d, e))) (A^2 -> H(e, U1(d, k)),A^2 -> H(e, U1(d, k))) ---------------------------------------- (2238) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(k, c)) A^2 -> H(e, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2239) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2240) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(b, e)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2241) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, e)),A^2 -> H(e, U1(c, e))) (A^2 -> H(e, U1(d, e)),A^2 -> H(e, U1(d, e))) ---------------------------------------- (2242) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(b, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2243) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(c, k)),A^2 -> H(e, U1(c, k))) (A^2 -> H(e, U1(d, k)),A^2 -> H(e, U1(d, k))) ---------------------------------------- (2244) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(e, c)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2245) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, c),A^2 -> H(e, c)) (A^2 -> H(e, U1(e, e)),A^2 -> H(e, U1(e, e))) (A^2 -> H(e, U1(e, k)),A^2 -> H(e, U1(e, k))) ---------------------------------------- (2246) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(c, e)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2247) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, e)),A^2 -> H(e, U1(e, e))) (A^2 -> H(e, U1(k, e)),A^2 -> H(e, U1(k, e))) ---------------------------------------- (2248) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2249) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2250) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(c, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2251) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, k)),A^2 -> H(e, U1(e, k))) (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) ---------------------------------------- (2252) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2253) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2254) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(e, e)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2255) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) ---------------------------------------- (2256) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(e, U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2257) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) ---------------------------------------- (2258) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(e, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2259) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2260) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), b) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2261) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, b),A^2 -> H(e, b)) (A^2 -> H(U1(e, e), c),A^2 -> H(U1(e, e), c)) (A^2 -> H(U1(e, e), d),A^2 -> H(U1(e, e), d)) ---------------------------------------- (2262) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), U1(e, d)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2263) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, d)),A^2 -> H(e, U1(e, d))) (A^2 -> H(U1(e, e), d),A^2 -> H(U1(e, e), d)) (A^2 -> H(U1(e, e), U1(e, k)),A^2 -> H(U1(e, e), U1(e, k))) ---------------------------------------- (2264) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), U1(d, e)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2265) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, e)),A^2 -> H(e, U1(d, e))) (A^2 -> H(U1(e, e), U1(k, e)),A^2 -> H(U1(e, e), U1(k, e))) ---------------------------------------- (2266) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2267) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, c),A^2 -> H(e, c)) (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) ---------------------------------------- (2268) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), U1(e, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2269) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, k)),A^2 -> H(e, U1(e, k))) (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) ---------------------------------------- (2270) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), U1(k, e)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2271) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, e)),A^2 -> H(e, U1(k, e))) ---------------------------------------- (2272) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(e, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2273) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2274) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), U1(k, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2275) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) ---------------------------------------- (2276) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(e, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2277) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2278) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(e), d) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2279) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), d),A^2 -> H(U1(e, e), d)) (A^2 -> H(f(e), k),A^2 -> H(f(e), k)) ---------------------------------------- (2280) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(e), k) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2281) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(e), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) ---------------------------------------- (2282) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2283) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) ---------------------------------------- (2284) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2285) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) ---------------------------------------- (2286) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2287) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2288) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2289) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2290) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2291) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2292) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), e) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2293) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2294) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2295) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2296) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2297) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2298) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(k), d) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2299) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(f(k), k),A^2 -> H(f(k), k)) ---------------------------------------- (2300) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(f(k), k) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2301) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(f(k), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2302) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2303) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2304) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, b)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2305) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(U1(k, k), U1(e, c)),A^2 -> H(U1(k, k), U1(e, c))) (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) ---------------------------------------- (2306) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(k, b)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2307) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) ---------------------------------------- (2308) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(c, d)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2309) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(U1(k, k), U1(k, d)),A^2 -> H(U1(k, k), U1(k, d))) (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) ---------------------------------------- (2310) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(d, c)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2311) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, c)),A^2 -> H(U1(k, k), U1(k, c))) (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) ---------------------------------------- (2312) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(b, e)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2313) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, e)),A^2 -> H(U1(k, k), U1(c, e))) (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) ---------------------------------------- (2314) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(b, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2315) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(c, k)),A^2 -> H(U1(k, k), U1(c, k))) (A^2 -> H(U1(k, k), U1(d, k)),A^2 -> H(U1(k, k), U1(d, k))) ---------------------------------------- (2316) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, c)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2317) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) ---------------------------------------- (2318) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(k, c)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2319) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2320) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2321) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2322) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(c, e)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2323) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2324) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2325) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2326) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(c, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2327) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2328) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2329) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (2330) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2331) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2332) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(k, d)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2333) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2334) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2335) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2336) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), b) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2337) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(U1(k, d), c),A^2 -> H(U1(k, d), c)) (A^2 -> H(U1(k, d), d),A^2 -> H(U1(k, d), d)) ---------------------------------------- (2338) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(e, d)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2339) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(U1(k, d), d),A^2 -> H(U1(k, d), d)) (A^2 -> H(U1(k, d), U1(e, k)),A^2 -> H(U1(k, d), U1(e, k))) ---------------------------------------- (2340) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(d, e)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2341) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(U1(k, d), U1(k, e)),A^2 -> H(U1(k, d), U1(k, e))) ---------------------------------------- (2342) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), c) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2343) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(k, d), e),A^2 -> H(U1(k, d), e)) (A^2 -> H(U1(k, d), k),A^2 -> H(U1(k, d), k)) ---------------------------------------- (2344) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(k, d), e) A^2 -> H(U1(k, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2345) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2346) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(e, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2347) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(U1(k, d), k),A^2 -> H(U1(k, d), k)) ---------------------------------------- (2348) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(k, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2349) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2350) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(k, e)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2351) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2352) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2353) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2354) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, d), U1(k, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2355) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2356) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), b) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2357) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), b),A^2 -> H(U1(k, k), b)) (A^2 -> H(U1(d, k), c),A^2 -> H(U1(d, k), c)) (A^2 -> H(U1(d, k), d),A^2 -> H(U1(d, k), d)) ---------------------------------------- (2358) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), U1(e, d)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2359) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, d)),A^2 -> H(U1(k, k), U1(e, d))) (A^2 -> H(U1(d, k), d),A^2 -> H(U1(d, k), d)) (A^2 -> H(U1(d, k), U1(e, k)),A^2 -> H(U1(d, k), U1(e, k))) ---------------------------------------- (2360) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), U1(d, e)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2361) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(d, e)),A^2 -> H(U1(k, k), U1(d, e))) (A^2 -> H(U1(d, k), U1(k, e)),A^2 -> H(U1(d, k), U1(k, e))) ---------------------------------------- (2362) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), c) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2363) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(d, k), e),A^2 -> H(U1(d, k), e)) (A^2 -> H(U1(d, k), k),A^2 -> H(U1(d, k), k)) ---------------------------------------- (2364) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), U1(e, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2365) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) (A^2 -> H(U1(d, k), k),A^2 -> H(U1(d, k), k)) ---------------------------------------- (2366) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), U1(k, e)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2367) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2368) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2369) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2370) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), e) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2371) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (2372) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2373) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2374) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, k), U1(k, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2375) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, k)),A^2 -> H(U1(k, k), U1(k, k))) ---------------------------------------- (2376) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), d) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2377) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), d),A^2 -> H(U1(k, d), d)) (A^2 -> H(U1(d, k), d),A^2 -> H(U1(d, k), d)) (A^2 -> H(U1(d, d), k),A^2 -> H(U1(d, d), k)) ---------------------------------------- (2378) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(d, d), k) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2379) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, d), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, d), k),A^2 -> H(U1(k, d), k)) (A^2 -> H(U1(d, k), k),A^2 -> H(U1(d, k), k)) ---------------------------------------- (2380) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(k, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2381) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2382) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), c) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2383) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, c),A^2 -> H(c, c)) (A^2 -> H(U1(e, e), c),A^2 -> H(U1(e, e), c)) (A^2 -> H(U1(e, k), c),A^2 -> H(U1(e, k), c)) (A^2 -> H(U1(e, c), e),A^2 -> H(U1(e, c), e)) (A^2 -> H(U1(e, c), k),A^2 -> H(U1(e, c), k)) ---------------------------------------- (2384) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), e) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2385) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), e),A^2 -> H(U1(e, c), e)) (A^2 -> H(U1(k, c), e),A^2 -> H(U1(k, c), e)) (A^2 -> H(U1(c, e), e),A^2 -> H(U1(c, e), e)) (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) ---------------------------------------- (2386) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(k, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2387) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2388) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, c), k) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2389) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, c), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, c), k),A^2 -> H(U1(e, c), k)) (A^2 -> H(U1(k, c), k),A^2 -> H(U1(k, c), k)) (A^2 -> H(U1(c, e), k),A^2 -> H(U1(c, e), k)) (A^2 -> H(U1(c, k), k),A^2 -> H(U1(c, k), k)) ---------------------------------------- (2390) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(U1(k, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2391) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2392) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, c), U1(e, e)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2393) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, e)),A^2 -> H(c, U1(e, e))) (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(e, c), e),A^2 -> H(U1(e, c), e)) ---------------------------------------- (2394) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(k, c), U1(e, e)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2395) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(k, c), e),A^2 -> H(U1(k, c), e)) ---------------------------------------- (2396) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(U1(k, c), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2397) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2398) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, e), U1(e, e)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2399) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), U1(e, e)),A^2 -> H(U1(e, e), U1(e, e))) (A^2 -> H(U1(k, e), U1(e, e)),A^2 -> H(U1(k, e), U1(e, e))) (A^2 -> H(U1(c, e), e),A^2 -> H(U1(c, e), e)) ---------------------------------------- (2400) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(c, k), U1(e, e)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2401) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), U1(e, e)),A^2 -> H(U1(e, k), U1(e, e))) (A^2 -> H(U1(k, k), U1(e, e)),A^2 -> H(U1(k, k), U1(e, e))) (A^2 -> H(U1(c, k), e),A^2 -> H(U1(c, k), e)) ---------------------------------------- (2402) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, e), e) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2403) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) ---------------------------------------- (2404) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(e, b)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2405) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, b)),A^2 -> H(k, U1(e, b))) (A^2 -> H(d, b),A^2 -> H(d, b)) (A^2 -> H(d, U1(e, c)),A^2 -> H(d, U1(e, c))) (A^2 -> H(d, U1(e, d)),A^2 -> H(d, U1(e, d))) ---------------------------------------- (2406) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(k, b)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2407) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(k, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, b)),A^2 -> H(k, U1(k, b))) (A^2 -> H(d, U1(k, c)),A^2 -> H(d, U1(k, c))) (A^2 -> H(d, U1(k, d)),A^2 -> H(d, U1(k, d))) ---------------------------------------- (2408) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(k, U1(k, b)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2409) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2410) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, d)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2411) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, d)),A^2 -> H(k, U1(c, d))) (A^2 -> H(d, U1(e, d)),A^2 -> H(d, U1(e, d))) (A^2 -> H(d, U1(k, d)),A^2 -> H(d, U1(k, d))) (A^2 -> H(d, U1(c, k)),A^2 -> H(d, U1(c, k))) ---------------------------------------- (2412) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(d, c)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2413) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, c)),A^2 -> H(k, U1(d, c))) (A^2 -> H(d, U1(k, c)),A^2 -> H(d, U1(k, c))) (A^2 -> H(d, U1(d, e)),A^2 -> H(d, U1(d, e))) (A^2 -> H(d, U1(d, k)),A^2 -> H(d, U1(d, k))) ---------------------------------------- (2414) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(b, e)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2415) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, e)),A^2 -> H(k, U1(b, e))) (A^2 -> H(d, U1(c, e)),A^2 -> H(d, U1(c, e))) (A^2 -> H(d, U1(d, e)),A^2 -> H(d, U1(d, e))) ---------------------------------------- (2416) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(b, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2417) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(b, k)),A^2 -> H(k, U1(b, k))) (A^2 -> H(d, U1(c, k)),A^2 -> H(d, U1(c, k))) (A^2 -> H(d, U1(d, k)),A^2 -> H(d, U1(d, k))) ---------------------------------------- (2418) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(e, c)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2419) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, c)),A^2 -> H(k, U1(e, c))) (A^2 -> H(d, c),A^2 -> H(d, c)) (A^2 -> H(d, U1(e, e)),A^2 -> H(d, U1(e, e))) (A^2 -> H(d, U1(e, k)),A^2 -> H(d, U1(e, k))) ---------------------------------------- (2420) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(k, c)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2421) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(k, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, c)),A^2 -> H(k, U1(k, c))) (A^2 -> H(d, U1(k, e)),A^2 -> H(d, U1(k, e))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (2422) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(k, U1(k, c)) A^2 -> H(d, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2423) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2424) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, e)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2425) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, e)),A^2 -> H(k, U1(c, e))) (A^2 -> H(d, U1(e, e)),A^2 -> H(d, U1(e, e))) (A^2 -> H(d, U1(k, e)),A^2 -> H(d, U1(k, e))) ---------------------------------------- (2426) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(c, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2427) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, k)),A^2 -> H(k, U1(c, k))) (A^2 -> H(d, U1(e, k)),A^2 -> H(d, U1(e, k))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (2428) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(e, e)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2429) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(d, e),A^2 -> H(d, e)) ---------------------------------------- (2430) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(k, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2431) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2432) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2433) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2434) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(k, d)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2435) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(k, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, d)),A^2 -> H(k, U1(k, d))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (2436) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(k, U1(k, d)) A^2 -> H(d, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2437) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2438) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(d, U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2439) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, k)),A^2 -> H(k, U1(d, k))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (2440) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, b) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2441) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, b),A^2 -> H(c, b)) (A^2 -> H(d, b),A^2 -> H(d, b)) (A^2 -> H(a, c),A^2 -> H(a, c)) (A^2 -> H(a, d),A^2 -> H(a, d)) ---------------------------------------- (2442) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(e, d)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2443) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, d)),A^2 -> H(c, U1(e, d))) (A^2 -> H(d, U1(e, d)),A^2 -> H(d, U1(e, d))) (A^2 -> H(a, d),A^2 -> H(a, d)) (A^2 -> H(a, U1(e, k)),A^2 -> H(a, U1(e, k))) ---------------------------------------- (2444) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(d, e)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2445) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(d, e)),A^2 -> H(c, U1(d, e))) (A^2 -> H(d, U1(d, e)),A^2 -> H(d, U1(d, e))) (A^2 -> H(a, U1(k, e)),A^2 -> H(a, U1(k, e))) ---------------------------------------- (2446) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, c) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2447) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, c),A^2 -> H(c, c)) (A^2 -> H(d, c),A^2 -> H(d, c)) (A^2 -> H(a, e),A^2 -> H(a, e)) (A^2 -> H(a, k),A^2 -> H(a, k)) ---------------------------------------- (2448) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(e, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2449) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(e, k)),A^2 -> H(c, U1(e, k))) (A^2 -> H(d, U1(e, k)),A^2 -> H(d, U1(e, k))) (A^2 -> H(a, k),A^2 -> H(a, k)) ---------------------------------------- (2450) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(k, e)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2451) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, e)),A^2 -> H(c, U1(k, e))) (A^2 -> H(d, U1(k, e)),A^2 -> H(d, U1(k, e))) ---------------------------------------- (2452) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, e) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2453) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, e),A^2 -> H(c, e)) (A^2 -> H(d, e),A^2 -> H(d, e)) ---------------------------------------- (2454) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(a, U1(k, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2455) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, U1(k, k)),A^2 -> H(c, U1(k, k))) (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) ---------------------------------------- (2456) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), b) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2457) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, b),A^2 -> H(d, b)) (A^2 -> H(U1(e, k), b),A^2 -> H(U1(e, k), b)) (A^2 -> H(U1(e, d), c),A^2 -> H(U1(e, d), c)) (A^2 -> H(U1(e, d), d),A^2 -> H(U1(e, d), d)) ---------------------------------------- (2458) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(d, e)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2459) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(d, e)),A^2 -> H(d, U1(d, e))) (A^2 -> H(U1(e, k), U1(d, e)),A^2 -> H(U1(e, k), U1(d, e))) (A^2 -> H(U1(e, d), U1(k, e)),A^2 -> H(U1(e, d), U1(k, e))) ---------------------------------------- (2460) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), c) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2461) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, c),A^2 -> H(d, c)) (A^2 -> H(U1(e, k), c),A^2 -> H(U1(e, k), c)) (A^2 -> H(U1(e, d), e),A^2 -> H(U1(e, d), e)) (A^2 -> H(U1(e, d), k),A^2 -> H(U1(e, d), k)) ---------------------------------------- (2462) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(e, k)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2463) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(e, k)),A^2 -> H(d, U1(e, k))) (A^2 -> H(U1(e, k), U1(e, k)),A^2 -> H(U1(e, k), U1(e, k))) (A^2 -> H(U1(e, d), k),A^2 -> H(U1(e, d), k)) ---------------------------------------- (2464) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(k, e)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2465) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(k, e)),A^2 -> H(d, U1(k, e))) (A^2 -> H(U1(e, k), U1(k, e)),A^2 -> H(U1(e, k), U1(k, e))) ---------------------------------------- (2466) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), e) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2467) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, e),A^2 -> H(d, e)) (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) ---------------------------------------- (2468) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, d), U1(k, k)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2469) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, U1(k, k)),A^2 -> H(d, U1(k, k))) (A^2 -> H(U1(e, k), U1(k, k)),A^2 -> H(U1(e, k), U1(k, k))) ---------------------------------------- (2470) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, a), d) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2471) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, d),A^2 -> H(a, d)) (A^2 -> H(U1(e, c), d),A^2 -> H(U1(e, c), d)) (A^2 -> H(U1(e, d), d),A^2 -> H(U1(e, d), d)) (A^2 -> H(U1(e, a), k),A^2 -> H(U1(e, a), k)) ---------------------------------------- (2472) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(e, a), k) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2473) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, a), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(a, k),A^2 -> H(a, k)) (A^2 -> H(U1(e, c), k),A^2 -> H(U1(e, c), k)) (A^2 -> H(U1(e, d), k),A^2 -> H(U1(e, d), k)) ---------------------------------------- (2474) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(k, a), d) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2475) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, a), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), d),A^2 -> H(U1(k, c), d)) (A^2 -> H(U1(k, d), d),A^2 -> H(U1(k, d), d)) (A^2 -> H(U1(k, a), k),A^2 -> H(U1(k, a), k)) ---------------------------------------- (2476) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(k, a), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2477) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2478) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(c, d), d) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2479) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), d),A^2 -> H(U1(e, d), d)) (A^2 -> H(U1(k, d), d),A^2 -> H(U1(k, d), d)) (A^2 -> H(U1(c, k), d),A^2 -> H(U1(c, k), d)) (A^2 -> H(U1(c, d), k),A^2 -> H(U1(c, d), k)) ---------------------------------------- (2480) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(c, d), k) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2481) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, d), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, d), k),A^2 -> H(U1(e, d), k)) (A^2 -> H(U1(k, d), k),A^2 -> H(U1(k, d), k)) (A^2 -> H(U1(c, k), k),A^2 -> H(U1(c, k), k)) ---------------------------------------- (2482) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(k, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2483) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2484) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), b) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2485) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), b),A^2 -> H(U1(k, e), b)) (A^2 -> H(U1(d, e), c),A^2 -> H(U1(d, e), c)) (A^2 -> H(U1(d, e), d),A^2 -> H(U1(d, e), d)) ---------------------------------------- (2486) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2487) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, d)),A^2 -> H(U1(k, e), U1(e, d))) (A^2 -> H(U1(d, e), d),A^2 -> H(U1(d, e), d)) (A^2 -> H(U1(d, e), U1(e, k)),A^2 -> H(U1(d, e), U1(e, k))) ---------------------------------------- (2488) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), c) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2489) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), c),A^2 -> H(U1(k, e), c)) (A^2 -> H(U1(d, e), e),A^2 -> H(U1(d, e), e)) (A^2 -> H(U1(d, e), k),A^2 -> H(U1(d, e), k)) ---------------------------------------- (2490) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), U1(e, k)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2491) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(e, k)),A^2 -> H(U1(k, e), U1(e, k))) (A^2 -> H(U1(d, e), k),A^2 -> H(U1(d, e), k)) ---------------------------------------- (2492) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), U1(k, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2493) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, e)),A^2 -> H(U1(k, e), U1(k, e))) ---------------------------------------- (2494) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), e) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2495) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), e),A^2 -> H(U1(k, e), e)) ---------------------------------------- (2496) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(U1(k, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2497) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2498) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, e), U1(k, k)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2499) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, k)),A^2 -> H(U1(k, e), U1(k, k))) ---------------------------------------- (2500) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(U1(k, e), U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2501) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2502) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, c), d) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2503) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), d),A^2 -> H(U1(k, c), d)) (A^2 -> H(U1(d, e), d),A^2 -> H(U1(d, e), d)) (A^2 -> H(U1(d, k), d),A^2 -> H(U1(d, k), d)) (A^2 -> H(U1(d, c), k),A^2 -> H(U1(d, c), k)) ---------------------------------------- (2504) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(d, c), k) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2505) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, c), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, c), k),A^2 -> H(U1(k, c), k)) (A^2 -> H(U1(d, e), k),A^2 -> H(U1(d, e), k)) (A^2 -> H(U1(d, k), k),A^2 -> H(U1(d, k), k)) ---------------------------------------- (2506) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(U1(k, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2507) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2508) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(a, e), d) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2509) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), d),A^2 -> H(U1(c, e), d)) (A^2 -> H(U1(d, e), d),A^2 -> H(U1(d, e), d)) (A^2 -> H(U1(a, e), k),A^2 -> H(U1(a, e), k)) ---------------------------------------- (2510) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(a, e), k) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2511) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, e), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, e), k),A^2 -> H(U1(c, e), k)) (A^2 -> H(U1(d, e), k),A^2 -> H(U1(d, e), k)) ---------------------------------------- (2512) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(a, k), d) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2513) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), d),A^2 -> H(U1(c, k), d)) (A^2 -> H(U1(d, k), d),A^2 -> H(U1(d, k), d)) (A^2 -> H(U1(a, k), k),A^2 -> H(U1(a, k), k)) ---------------------------------------- (2514) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(a, k), k) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2515) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(a, k), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(c, k), k),A^2 -> H(U1(c, k), k)) (A^2 -> H(U1(d, k), k),A^2 -> H(U1(d, k), k)) ---------------------------------------- (2516) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(e, b)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2517) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(e, b)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, b),A^2 -> H(k, b)) (A^2 -> H(k, U1(e, c)),A^2 -> H(k, U1(e, c))) (A^2 -> H(k, U1(e, d)),A^2 -> H(k, U1(e, d))) ---------------------------------------- (2518) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(c, d)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2519) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(c, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, d)),A^2 -> H(k, U1(e, d))) (A^2 -> H(k, U1(k, d)),A^2 -> H(k, U1(k, d))) (A^2 -> H(k, U1(c, k)),A^2 -> H(k, U1(c, k))) ---------------------------------------- (2520) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(k, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2521) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2522) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(d, c)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2523) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(d, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, c)),A^2 -> H(k, U1(k, c))) (A^2 -> H(k, U1(d, e)),A^2 -> H(k, U1(d, e))) (A^2 -> H(k, U1(d, k)),A^2 -> H(k, U1(d, k))) ---------------------------------------- (2524) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(k, c)) A^2 -> H(k, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2525) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2526) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(b, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2527) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(b, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, e)),A^2 -> H(k, U1(c, e))) (A^2 -> H(k, U1(d, e)),A^2 -> H(k, U1(d, e))) ---------------------------------------- (2528) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(b, k)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2529) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(b, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(c, k)),A^2 -> H(k, U1(c, k))) (A^2 -> H(k, U1(d, k)),A^2 -> H(k, U1(d, k))) ---------------------------------------- (2530) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(e, c)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2531) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(e, c)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, c),A^2 -> H(k, c)) (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(k, U1(e, k)),A^2 -> H(k, U1(e, k))) ---------------------------------------- (2532) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(c, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2533) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(c, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(k, U1(k, e)),A^2 -> H(k, U1(k, e))) ---------------------------------------- (2534) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(k, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2535) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2536) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(c, k)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2537) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(c, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, k)),A^2 -> H(k, U1(e, k))) (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2538) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2539) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2540) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2541) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2542) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2543) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2544) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(k, U1(d, k)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2545) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(d, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2546) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2547) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2548) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, b) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2549) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, b),A^2 -> H(e, b)) (A^2 -> H(k, b),A^2 -> H(k, b)) (A^2 -> H(c, c),A^2 -> H(c, c)) (A^2 -> H(c, d),A^2 -> H(c, d)) ---------------------------------------- (2550) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, U1(e, d)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2551) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, d)),A^2 -> H(e, U1(e, d))) (A^2 -> H(k, U1(e, d)),A^2 -> H(k, U1(e, d))) (A^2 -> H(c, d),A^2 -> H(c, d)) (A^2 -> H(c, U1(e, k)),A^2 -> H(c, U1(e, k))) ---------------------------------------- (2552) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2553) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(d, e)),A^2 -> H(e, U1(d, e))) (A^2 -> H(k, U1(d, e)),A^2 -> H(k, U1(d, e))) (A^2 -> H(c, U1(k, e)),A^2 -> H(c, U1(k, e))) ---------------------------------------- (2554) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(e, k)) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2555) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, k)),A^2 -> H(e, U1(e, k))) (A^2 -> H(k, U1(e, k)),A^2 -> H(k, U1(e, k))) (A^2 -> H(c, k),A^2 -> H(c, k)) ---------------------------------------- (2556) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(k, e)) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2557) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, e)),A^2 -> H(e, U1(k, e))) (A^2 -> H(k, U1(k, e)),A^2 -> H(k, U1(k, e))) ---------------------------------------- (2558) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(e, U1(k, e)) A^2 -> H(k, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2559) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2560) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, e) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2561) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2562) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2563) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2564) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(c, U1(k, k)) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2565) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, k)),A^2 -> H(e, U1(k, k))) (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2566) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(e, U1(k, k)) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2567) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2568) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), b) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2569) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, b),A^2 -> H(k, b)) (A^2 -> H(U1(e, k), c),A^2 -> H(U1(e, k), c)) (A^2 -> H(U1(e, k), d),A^2 -> H(U1(e, k), d)) ---------------------------------------- (2570) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, d)) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2571) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, d)),A^2 -> H(k, U1(e, d))) (A^2 -> H(U1(e, k), d),A^2 -> H(U1(e, k), d)) (A^2 -> H(U1(e, k), U1(e, k)),A^2 -> H(U1(e, k), U1(e, k))) ---------------------------------------- (2572) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(d, e)) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2573) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, e)),A^2 -> H(k, U1(d, e))) (A^2 -> H(U1(e, k), U1(k, e)),A^2 -> H(U1(e, k), U1(k, e))) ---------------------------------------- (2574) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2575) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, c),A^2 -> H(k, c)) (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) (A^2 -> H(U1(e, k), k),A^2 -> H(U1(e, k), k)) ---------------------------------------- (2576) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, e)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2577) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, e)),A^2 -> H(k, U1(k, e))) ---------------------------------------- (2578) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(k, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2579) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2580) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), e) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2581) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2582) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2583) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2584) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, k), U1(k, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2585) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2586) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2587) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2588) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, c), d) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2589) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, d),A^2 -> H(c, d)) (A^2 -> H(U1(e, e), d),A^2 -> H(U1(e, e), d)) (A^2 -> H(U1(e, k), d),A^2 -> H(U1(e, k), d)) (A^2 -> H(U1(e, c), k),A^2 -> H(U1(e, c), k)) ---------------------------------------- (2590) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(e, c), k) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2591) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, k),A^2 -> H(c, k)) (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) (A^2 -> H(U1(e, k), k),A^2 -> H(U1(e, k), k)) ---------------------------------------- (2592) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), b) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2593) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), c),A^2 -> H(U1(k, e), c)) (A^2 -> H(U1(k, e), d),A^2 -> H(U1(k, e), d)) ---------------------------------------- (2594) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(e, d)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2595) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), d),A^2 -> H(U1(k, e), d)) (A^2 -> H(U1(k, e), U1(e, k)),A^2 -> H(U1(k, e), U1(e, k))) ---------------------------------------- (2596) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(d, e)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2597) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), U1(k, e)),A^2 -> H(U1(k, e), U1(k, e))) ---------------------------------------- (2598) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), c) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2599) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), e),A^2 -> H(U1(k, e), e)) (A^2 -> H(U1(k, e), k),A^2 -> H(U1(k, e), k)) ---------------------------------------- (2600) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(U1(k, e), e) A^2 -> H(U1(k, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2601) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2602) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2603) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), k),A^2 -> H(U1(k, e), k)) ---------------------------------------- (2604) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(U1(k, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2605) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2606) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(k, c), d) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2607) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, c), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), d),A^2 -> H(U1(k, e), d)) (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(U1(k, c), k),A^2 -> H(U1(k, c), k)) ---------------------------------------- (2608) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(U1(k, c), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2609) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2610) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(c, e), d) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2611) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), d),A^2 -> H(U1(e, e), d)) (A^2 -> H(U1(k, e), d),A^2 -> H(U1(k, e), d)) (A^2 -> H(U1(c, e), k),A^2 -> H(U1(c, e), k)) ---------------------------------------- (2612) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(c, e), k) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2613) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) (A^2 -> H(U1(k, e), k),A^2 -> H(U1(k, e), k)) ---------------------------------------- (2614) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(U1(k, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2615) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2616) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(c, k), d) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2617) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), d),A^2 -> H(U1(e, k), d)) (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(U1(c, k), k),A^2 -> H(U1(c, k), k)) ---------------------------------------- (2618) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(U1(c, k), k) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2619) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), k),A^2 -> H(U1(e, k), k)) (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2620) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2621) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2622) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, b) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2623) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, c),A^2 -> H(e, c)) (A^2 -> H(e, d),A^2 -> H(e, d)) ---------------------------------------- (2624) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, U1(e, d)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2625) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, d),A^2 -> H(e, d)) (A^2 -> H(e, U1(e, k)),A^2 -> H(e, U1(e, k))) ---------------------------------------- (2626) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, U1(d, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2627) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(k, e)),A^2 -> H(e, U1(k, e))) ---------------------------------------- (2628) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(e, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2629) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2630) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, c) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2631) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) (A^2 -> H(e, k),A^2 -> H(e, k)) ---------------------------------------- (2632) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2633) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2634) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, U1(e, k)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2635) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, k),A^2 -> H(e, k)) ---------------------------------------- (2636) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2637) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2638) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), d) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2639) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, d),A^2 -> H(e, d)) (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) ---------------------------------------- (2640) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), k) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2641) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, k),A^2 -> H(e, k)) ---------------------------------------- (2642) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2643) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2644) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2645) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2646) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2647) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2648) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), b) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2649) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), c),A^2 -> H(U1(k, k), c)) (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) ---------------------------------------- (2650) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), U1(e, d)) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2651) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(U1(k, k), U1(e, k)),A^2 -> H(U1(k, k), U1(e, k))) ---------------------------------------- (2652) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), U1(d, e)) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2653) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), U1(k, e)),A^2 -> H(U1(k, k), U1(k, e))) ---------------------------------------- (2654) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2655) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2656) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), c) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2657) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2658) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), e) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2659) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (2660) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), U1(e, k)) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2661) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2662) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2663) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2664) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, d), d) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2665) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, d), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(U1(k, d), k),A^2 -> H(U1(k, d), k)) ---------------------------------------- (2666) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, d), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2667) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2668) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(d, k), d) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2669) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), d),A^2 -> H(U1(k, k), d)) (A^2 -> H(U1(d, k), k),A^2 -> H(U1(d, k), k)) ---------------------------------------- (2670) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(d, k), k) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2671) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, k), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2672) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2673) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2674) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), c) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2675) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, c),A^2 -> H(e, c)) (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) (A^2 -> H(U1(e, e), k),A^2 -> H(U1(e, e), k)) ---------------------------------------- (2676) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, c), e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2677) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, c), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, e),A^2 -> H(c, e)) (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) ---------------------------------------- (2678) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(c, e), e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2679) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, e), e),A^2 -> H(U1(e, e), e)) (A^2 -> H(U1(k, e), e),A^2 -> H(U1(k, e), e)) ---------------------------------------- (2680) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(U1(k, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2681) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2682) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(c, k), e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2683) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(c, k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (2684) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2685) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2686) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(c, U1(e, e)) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2687) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, U1(e, e)),A^2 -> H(e, U1(e, e))) (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(c, e),A^2 -> H(c, e)) ---------------------------------------- (2688) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, k), U1(e, e)) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2689) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, e)),A^2 -> H(k, U1(e, e))) (A^2 -> H(U1(e, k), e),A^2 -> H(U1(e, k), e)) ---------------------------------------- (2690) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, e), U1(e, e)) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2691) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), e),A^2 -> H(U1(k, e), e)) ---------------------------------------- (2692) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(U1(k, e), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2693) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2694) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), U1(e, e)) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2695) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), e),A^2 -> H(U1(k, k), e)) ---------------------------------------- (2696) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(U1(k, k), e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2697) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2698) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, b) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2699) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, b),A^2 -> H(k, b)) (A^2 -> H(d, c),A^2 -> H(d, c)) (A^2 -> H(d, d),A^2 -> H(d, d)) ---------------------------------------- (2700) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, U1(e, d)) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2701) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, d)),A^2 -> H(k, U1(e, d))) (A^2 -> H(d, d),A^2 -> H(d, d)) (A^2 -> H(d, U1(e, k)),A^2 -> H(d, U1(e, k))) ---------------------------------------- (2702) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, U1(d, e)) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2703) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(d, e)),A^2 -> H(k, U1(d, e))) (A^2 -> H(d, U1(k, e)),A^2 -> H(d, U1(k, e))) ---------------------------------------- (2704) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, c) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2705) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, c),A^2 -> H(k, c)) (A^2 -> H(d, e),A^2 -> H(d, e)) (A^2 -> H(d, k),A^2 -> H(d, k)) ---------------------------------------- (2706) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, U1(e, k)) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2707) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(e, k)),A^2 -> H(k, U1(e, k))) (A^2 -> H(d, k),A^2 -> H(d, k)) ---------------------------------------- (2708) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, U1(k, e)) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2709) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(k, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, e)),A^2 -> H(k, U1(k, e))) ---------------------------------------- (2710) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2711) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2712) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2713) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2714) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2715) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2716) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, U1(k, k)) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2717) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, U1(k, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, k)),A^2 -> H(k, U1(k, k))) ---------------------------------------- (2718) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, U1(k, k)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2719) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2720) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(a, d) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2721) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, d),A^2 -> H(c, d)) (A^2 -> H(d, d),A^2 -> H(d, d)) (A^2 -> H(a, k),A^2 -> H(a, k)) ---------------------------------------- (2722) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(a, k) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2723) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(a, k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(c, k),A^2 -> H(c, k)) (A^2 -> H(d, k),A^2 -> H(d, k)) ---------------------------------------- (2724) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, d), d) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2725) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, d),A^2 -> H(d, d)) (A^2 -> H(U1(e, k), d),A^2 -> H(U1(e, k), d)) (A^2 -> H(U1(e, d), k),A^2 -> H(U1(e, d), k)) ---------------------------------------- (2726) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, d), k) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2727) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, d), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(d, k),A^2 -> H(d, k)) (A^2 -> H(U1(e, k), k),A^2 -> H(U1(e, k), k)) ---------------------------------------- (2728) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(d, e), d) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2729) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), d),A^2 -> H(U1(k, e), d)) (A^2 -> H(U1(d, e), k),A^2 -> H(U1(d, e), k)) ---------------------------------------- (2730) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(d, e), k) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2731) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(d, e), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), k),A^2 -> H(U1(k, e), k)) ---------------------------------------- (2732) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(U1(k, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2733) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2734) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, b) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2735) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, b) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, c),A^2 -> H(k, c)) (A^2 -> H(k, d),A^2 -> H(k, d)) ---------------------------------------- (2736) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, U1(e, d)) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2737) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(e, d)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, d),A^2 -> H(k, d)) (A^2 -> H(k, U1(e, k)),A^2 -> H(k, U1(e, k))) ---------------------------------------- (2738) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, U1(d, e)) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2739) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(d, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, U1(k, e)),A^2 -> H(k, U1(k, e))) ---------------------------------------- (2740) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, U1(k, e)) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2741) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2742) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, c) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2743) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, e),A^2 -> H(k, e)) (A^2 -> H(k, k),A^2 -> H(k, k)) ---------------------------------------- (2744) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, e) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2745) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2746) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, U1(e, k)) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2747) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(e, k)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, k),A^2 -> H(k, k)) ---------------------------------------- (2748) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(c, d) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2749) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, d),A^2 -> H(e, d)) (A^2 -> H(k, d),A^2 -> H(k, d)) (A^2 -> H(c, k),A^2 -> H(c, k)) ---------------------------------------- (2750) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(c, k) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2751) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, k),A^2 -> H(e, k)) (A^2 -> H(k, k),A^2 -> H(k, k)) ---------------------------------------- (2752) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2753) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2754) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, k), d) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2755) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, d),A^2 -> H(k, d)) (A^2 -> H(U1(e, k), k),A^2 -> H(U1(e, k), k)) ---------------------------------------- (2756) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, k), k) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2757) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, k),A^2 -> H(k, k)) ---------------------------------------- (2758) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, e), d) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2759) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, e), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, e), k),A^2 -> H(U1(k, e), k)) ---------------------------------------- (2760) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(U1(k, e), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2761) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2762) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(e, d) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2763) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, k),A^2 -> H(e, k)) ---------------------------------------- (2764) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2765) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2766) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(k, k), d) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2767) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(k, k), d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(U1(k, k), k),A^2 -> H(U1(k, k), k)) ---------------------------------------- (2768) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(U1(k, k), k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2769) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2770) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(e, c) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2771) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, c) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) (A^2 -> H(e, k),A^2 -> H(e, k)) ---------------------------------------- (2772) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2773) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2774) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2775) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) ---------------------------------------- (2776) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, e), k) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2777) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, e), k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, k),A^2 -> H(e, k)) ---------------------------------------- (2778) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(e, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2779) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2780) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(c, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2781) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(c, e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2782) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2783) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2784) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(U1(e, k), e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2785) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(U1(e, k), e) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2786) Obligation: Q DP problem: The TRS P consists of the following rules: G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2787) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2788) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(e, U1(e, e)) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2789) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(e, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(e, e),A^2 -> H(e, e)) ---------------------------------------- (2790) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(k, U1(e, e)) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2791) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, U1(e, e)) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, e),A^2 -> H(k, e)) ---------------------------------------- (2792) Obligation: Q DP problem: The TRS P consists of the following rules: A^2 -> H(f(c), f(c)) H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) A^2 -> H(k, e) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2793) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (2794) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, d) A^2 -> H(d, k) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2795) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(d, k) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, k),A^2 -> H(k, k)) ---------------------------------------- (2796) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, d) A^2 -> H(k, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2797) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule A^2 -> H(k, d) at position [] we obtained the following new rules [LPAR04]: (A^2 -> H(k, k),A^2 -> H(k, k)) ---------------------------------------- (2798) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, d) A^2 -> H(k, k) The TRS R consists of the following rules: f(x) -> U1(x, x) U1(e, x) -> x a -> c a -> d b -> c b -> d d -> k c -> e c -> k Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2799) UsableRulesReductionPairsProof (EQUIVALENT) By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well. No dependency pairs are removed. The following rules are removed from R: a -> c a -> d b -> c b -> d Used ordering: POLO with Polynomial interpretation [POLO]: POL(A^2) = 0 POL(G(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3 POL(H(x_1, x_2)) = x_1 + x_2 POL(U1(x_1, x_2)) = x_1 + x_2 POL(c) = 0 POL(d) = 0 POL(e) = 0 POL(f(x_1)) = 2*x_1 POL(k) = 0 ---------------------------------------- (2800) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(f(k), f(k)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, d) A^2 -> H(k, k) The TRS R consists of the following rules: d -> k U1(e, x) -> x c -> e c -> k f(x) -> U1(x, x) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2801) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A^2 -> H(f(k), f(k)) A^2 -> H(U1(c, k), U1(c, k)) A^2 -> H(U1(k, k), U1(k, k)) A^2 -> H(U1(d, k), U1(d, k)) A^2 -> H(U1(e, k), U1(e, k)) A^2 -> H(k, k) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. H(x1, x2) = x1 G(x1, x2, x3) = x1 f(x1) = x1 k = k d = d A^2 = A^2 c = c U1(x1, x2) = x2 e = e Recursive path order with status [RPO]. Quasi-Precedence: [d, A^2, c, e] > k Status: k: multiset status d: multiset status A^2: multiset status c: multiset status e: multiset status The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f(x) -> U1(x, x) c -> e c -> k d -> k U1(e, x) -> x ---------------------------------------- (2802) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(f(e), f(e)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(c, c) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) A^2 -> H(d, d) The TRS R consists of the following rules: d -> k U1(e, x) -> x c -> e c -> k f(x) -> U1(x, x) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2803) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A^2 -> H(f(e), f(e)) A^2 -> H(U1(c, e), U1(c, e)) A^2 -> H(U1(e, e), U1(e, e)) A^2 -> H(U1(d, e), U1(d, e)) A^2 -> H(U1(k, e), U1(k, e)) A^2 -> H(e, e) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. H(x1, x2) = x1 G(x1, x2, x3) = x1 f(x1) = x1 k = k d = d A^2 = A^2 c = c U1(x1, x2) = x2 e = e Recursive path order with status [RPO]. Quasi-Precedence: [d, A^2, c] > k [d, A^2, c] > e Status: k: multiset status d: multiset status A^2: multiset status c: multiset status e: multiset status The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f(x) -> U1(x, x) c -> e c -> k d -> k U1(e, x) -> x ---------------------------------------- (2804) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(c), f(c)) A^2 -> H(f(d), f(d)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(c, c) A^2 -> H(d, d) The TRS R consists of the following rules: d -> k U1(e, x) -> x c -> e c -> k f(x) -> U1(x, x) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2805) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. A^2 -> H(f(c), f(c)) A^2 -> H(U1(c, c), U1(c, c)) A^2 -> H(U1(d, c), U1(d, c)) A^2 -> H(U1(e, c), U1(e, c)) A^2 -> H(U1(k, c), U1(k, c)) A^2 -> H(c, c) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. H(x1, x2) = x2 G(x1, x2, x3) = x1 f(x1) = x1 k = k d = d A^2 = A^2 c = c U1(x1, x2) = x2 e = e Recursive path order with status [RPO]. Quasi-Precedence: [d, A^2] > [k, c] > e Status: k: multiset status d: multiset status A^2: multiset status c: multiset status e: multiset status The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: f(x) -> U1(x, x) c -> e c -> k d -> k U1(e, x) -> x ---------------------------------------- (2806) Obligation: Q DP problem: The TRS P consists of the following rules: H(x, x) -> G(x, x, f(k)) G(d, x, x) -> A^2 A^2 -> H(f(d), f(d)) A^2 -> H(U1(d, d), U1(d, d)) A^2 -> H(U1(c, d), U1(c, d)) A^2 -> H(U1(k, d), U1(k, d)) A^2 -> H(U1(e, d), U1(e, d)) A^2 -> H(d, d) The TRS R consists of the following rules: d -> k U1(e, x) -> x c -> e c -> k f(x) -> U1(x, x) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (2807) NonTerminationLoopProof (COMPLETE) We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. Found a loop by narrowing to the left: s = G(U1(c, d), U1(c, d), f(k)) evaluates to t =G(U1(c, d), U1(c, d), f(k)) Thus s starts an infinite chain as s semiunifies with t with the following substitutions: * Matcher: [ ] * Semiunifier: [ ] -------------------------------------------------------------------------------- Rewriting sequence G(U1(c, d), U1(c, d), f(k)) -> G(U1(c, d), U1(c, k), f(k)) with rule d -> k at position [1,1] and matcher [ ] G(U1(c, d), U1(c, k), f(k)) -> G(U1(c, d), U1(k, k), f(k)) with rule c -> k at position [1,0] and matcher [ ] G(U1(c, d), U1(k, k), f(k)) -> G(U1(c, d), U1(k, k), U1(k, k)) with rule f(x') -> U1(x', x') at position [2] and matcher [x' / k] G(U1(c, d), U1(k, k), U1(k, k)) -> G(U1(e, d), U1(k, k), U1(k, k)) with rule c -> e at position [0,0] and matcher [ ] G(U1(e, d), U1(k, k), U1(k, k)) -> G(d, U1(k, k), U1(k, k)) with rule U1(e, x') -> x' at position [0] and matcher [x' / d] G(d, U1(k, k), U1(k, k)) -> A^2 with rule G(d, x, x) -> A^2 at position [] and matcher [x / U1(k, k)] A^2 -> H(U1(c, d), U1(c, d)) with rule A^2 -> H(U1(c, d), U1(c, d)) at position [] and matcher [ ] H(U1(c, d), U1(c, d)) -> G(U1(c, d), U1(c, d), f(k)) with rule H(x, x) -> G(x, x, f(k)) Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence All these steps are and every following step will be a correct step w.r.t to Q. ---------------------------------------- (2808) NO